A COMPUTATIONAL IMPLEMENTATION OF

DESIGN SENSITIVITY ANALYSIS AND

STRUCTURAL OPTIMISATION

A dissertation submitted to

The Department of MATHEMATICS and APPLIED MATHEMATICS

of the University of Cape Town

in fulfillment of the requirements for the degree of

MASTER OF SCIENCE

m

APPLIED NIATHEMATICS

By Andre Smith Bothma

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The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or non-commercial research purposes only.

Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author.

Summary

111 the field of computational mechanics, increases in computing power and enhance­ments in material and kinematic models have enhanced the feasibility of performing structural design optimisation for a wide range of applications. The work presented here was motivated by the current groundswell of research effort in computational optimisation.

Design Sensitivity Analysis (DSA) crucially underpins much of structural optimisa­tion and, as such, ·is focussed on more intently than the optimisation theory itself: various approaches to the Direct Differentiation Method (DDM) DSA procedure are investigated and computationally implemented. The procedures implemented were chosen so as to involve a range of important issues in computational sensitivity analysis, particularly

• Shape and non-shape sensitivity analysis,

• Total and Updated Lagrange-based DSA,

• DSA of displacement and non-displacement based response functionals,

• Multiparameter DSA.

• DSA for large strain behaviour

The primary objectives of this thesis are:

I. Development of a design sensitivity formulation which, when discretised, re­sembles the standard displacement based kinematic element formulation, thus enabling the implementation of design sensitivity analysis in established Finite Element Analysis (FEA) codes as a 'pseudo~-element routine.

II. lmplemention of several design sensitivity formulations and structural optimi­sation into the FEA code ABAQUS as a verification of the first objective.

Numerical results provided in this work demonstrate the successful completion of the above-mentioned objectives. The discretised DSA formulations presented, as

well as the ·pseudo'-element approach adopted, particularly in the case of shape DSA. are entirely original. To the best of the author's knowledge, DSA and DSA­based structural optimisation had never before been attempted with ABAQCS. The research conducted here lays the foundation for potentially very fruitful future work.

11

Acknowledgements

With any time consuming and labourious endeavour one inevitably accrues debts, financial and otherwise, for support provided. It is with full understanding of the complex set of conditions which have to be maintained in order to make work like this possible that I wish to express my gratitude

•.To Dr. Jacek Ronda (Dr.Sc.) for supervising this thesis.

• To the FRD (Foundation for Research Development) and CERECAM (FRD / UCT Centre for Research into Computational and Applied Mechanics) for the provision of financial support and computational facilities.

• To all the people who made the system work the way it should in times of need, particularly Mike Eastman and Prof. B.D. Reddy.

• To colleagues, comrades-in-arms and friends in the Mathematics and Applied Mathematics department, too numerous to mention: the last couple of years have been the most memorable yet.

• Finally, to Pam, for everything.

l1l

Contents

1 Introduction 1

2 Structural Analysis 15

2.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1. l First Principles: The displacement-based equilibrium formu-lation . . . . . . . . . . 15

2.1.2 Kinematic Nonlinearity . 18

2.2 Discretisation . . . . . . . . .. 25

3 Design Sensitivity Analysis 34

3.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 34

:3.1.1 General Ideas on the formulation of Design Sensitivity Analysis 34

3.1.2 'Intuitive' Approach

3.1.3 'Rigorous' Approach

3.1...! Stress Sensitivity Analysis

:3. 2 Discretisation . . . . . . . . . . .

3.2.l 'Intuitive' Non-shape DSA

:3.2.2 'Rigorous' Non-shape DSA .

3.2.3 'Rigorous' Shape DSA

:3.2.4 Stress DSA ..... .

IV

.. ..

38

38 (

42

43

43

44

45

47

4 Structural Optimisation 51

4.1 Structural Optimisation, Design Sensitivity and Finite Element Ana-lysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1. l Structural Optimisation with the PLBA RQP

4.1.2 The PLBA RQP method algorithm ..... .

5 Implementing DSA and Optimisation into ABAQUS

6 Numerical Results

6.1 The :3-bar truss

6.2 The 10-bar truss·

6.3 The 200-bar truss .

6.4 The 25-bar pylon truss

6.5 Optimising the 25-bar pylon truss

7 Conclusion

7. l Research Achievements .

7.2 Results Discussion ...

7.2.l Design Sensitivity Analysis .

.., ') ·) '·-·- Structural Optimisation

7. 3 Future Research . . . . . . . . .

v

52

58

59

67

67

75

84

95

. 109

114

. 114

. 115

. 115

. 116

. 117

Nomenclature

Abbreviation and Definition First Defined AVM/ASM Adjoint Variable/System Method p. 6 BEM Boundary Element Method p. 2 BFGS Broyden-Fletcher-Goldfarb-Shanno solution procedure p. 53 DDM Direct Differentiation Method p. 1 DSA Design Sensitivity Analysis p. 1 DSC Design Sensitivity Coefficient p. 4 FDM Finite Difference Method p. 6 FEA Finite Element Analysis p. 1 FEM Finite Element Method p. 1 MP Mathematical Programming p. 9 :\JEL Number of Elements p. lll NF Number of Function calls p. 110 NI Number of Iterations p. llO NLP Nonlinear Programming p. 11 N-R Newton-Raphson iterative solution procedure p. 21 PLBA Pshenichny-Lim-Belegundu-Arora ") p. -QP Quadratic Programming p. 53 QPP Quadratic Programming Problem p. 53 RQP Recursive Quadratic Programming p. 2 SQP Successive Quadratic Programming p. 11 TL Total Lagrange

p. 13 UL Updated Lagrange p. 13

In generating results for structural analysis, DSA and design optimisation, the units employed for the various example problems correspond to that employed in the source literature. The following table provides. the conversion factors.

-Imperial Unit Definition Converted to SI 1 kip 1000 lbf 4.4482xl03 N 1 ksi 1000 psi 6.8948xl03 kPa

1 inch 2.5400x10- 2 m

VI

i / ,.o

10

10 i+.a.i 0 t

O' 0"

n o(k)

ox.i:

01

'V 0 .1

bO

bO bO Jo

.4 Q

Cu' c,. 1c:, Ct C" 0

E f F (u, b) (])

9 <I>

ru f 1

11' 12 I~

J. Jr L p (b. µ) Q ( u. b) f)

(}

:R p s t

v w

Symbol and Definition Value of quantity 0 defined i.t.o. the parent frame Cr Value of quantity O at time t, defined i.t.o. frame tc Incremental quantity o defined i.t.o. frame tc Quantity O at t + .at, defined i.t.o. frame tc Quantity O defined i.t.o. frame C0 '

Quantity 0 defined i.t.o. frame c; Quantity n is shape design parameter, h, dependent kth iterative estimate of quantity D =Du

Jth element discretisation of quantity o· Gradient of quantity D w.r.t. cross-sectional area A Variation of expression D

Full design parameter differentiation operator

Explicit design parameter differentiation operator Implicit design parameter differentiation operator

Truss cross-sectional area Stepsize parameter Local. or principal, frame Natural, or parent, frame Configuration/frame at time t Unperturbed 'initial' global frame, cf. shape DSA Young's modulus Cost, or objective, functional Equilibrium expression Rotation angle defined in Fig.(2.4] Iterative rotation angle Rotational multiplicative factor Surface of prescribed displacement Surface of prescribed traction Constraint sets Active constraints set Jacobian integration factors Truss element length Unconstrained Lagrange function Internal load expression Rotation angle defined in Fig.(2.4r lterati ve rotation angle External load expression Material density Body surface area Time Body volume

General design parameter dependent functional Vll

First Defined/Used p. 26 p. 18 p. 19 p. 19 p. 26 p. 41 p. 41 p. 21 p. 25 p. 37 p. 43 p. 17 p. 36 p. 36 p. 36

p. 28 p. 5.5 p. 26 p. 26 p. 19 p. 41 p. 25 p. 34 p. 36

"r p. _(

p. 31 p. 32 p. 16 p. 16 p. 53 p. 53 p. 26 p. 26

... ') p. u-p. 36 p. 27 p. 31 p. IS p. 16 p. 16 p. 18 p. 17 p. 34

Symbol and Definition First Defined/Used I a Linear acceleration vector

p. 16 b Design parameter vector r p. _, b Optimal design parameter vector p. 55 c Material constitutive tensor p. 20 d Global truss nodal displacement vector ?" p. _(S e Infinitesimal strain tensor/ vector p. 17 e Green-Lagrange strain tensor/vector p. 18 f Body force vector p. 16 g Constraint functions vector p. 34 G 0

, Gi Cost and constraint function sensitivity vectors p .. 54 h Shape design parameter vector p. 41

T/ Nonlinear part of Green-Lagrange strain p. 20 tensor/ vector

]{ Stiffness matrix p. 29 J{' 'Pseudo'-stiffness matrix p. 44 }(, TL local frame internal load matrix p. 29 }(, = 'Vo }(, p. 43 .-\

C, UL local frame internal load matrix p. 33 C, =Vo AC, p. 43 µ Lagrange multipliers p . .55 n Body normal vector p. 16 N Shape function matrix p. 27

Noo Shape function derivative vector p. 28 ,

N Shape function derivative factor p. 28 N Shape function derivative factor p. 28 N Shape function derivative factor

p. 29 N Shape function derivative factor p. 32 p Vector direction of design parameter change p. 53 R' 'Pseudo'-external load vector p. 44 s

2nd Piola-Kirchhoff stress tensor/vector p. 18 t Traction load vector p. 16 T Coordinate transformation matrix p. 27 T Transformation matrix component p. 27 T Cauchy stress tensor /vector p. 16 u Displacement vector p. 17

Hessian matrix for P .~

p. ,57 w

~

a! Coordinate vector ~ p. 16 1+~1x = a 1+~ 1x,

p. 19 t •,J &1x)

x Global nodal coordinate vector of a truss element p. 27 x Coordinate transformation factor p. 47 X' Local nodal coordinate vector of a truss element p. 27 ~ Natural. or parent, frame coordinate vector p. 26

Chapter 1

Introduction

Background

In industry and manufacturing, structural design specifications commonly demand some manner of optimisation, e.g. for the purposes of material cost reduction or mass minimisation. The optimisation procedure is usually constrained in terms of structural response limits1 imposed for a whole range of realistic loci.ding conditions.

The range of approaches to the solution of Structur~ Optimisation problems, both computational and analytical, are many and varied. In the research presented here a computational structural optimisation procedure is formulated and implemented in conjunction with the Finite Element Analysis (FEA) code ABAQUS; the procedure consists of three discrete components:

• The kinematic and material formulation describes the structural behaviour subject to loads and boundary conditions. The Finite Element Method (FEM) is adopted as a numerical solution procedure.

• Design Sensitivity Analysis (DSA) solves for the. sensitivity matrices

dr de du \7 b T = db ' \7 be = db and \7 bu = db '

which express the influence of design parameters b on structural response. Most optimisation procedures rely cruciall~n the sensitivity matrices in order to obtain an optimal value. Varjous approaches to the Direct Differentiation Method (DDM) are utilised to perform both shape and non-shape sensitivity analysis.

1 Defined in terms of displacement, strain and stress.

1

2

• A nonlinear optimisation model, designed to interface with ABAQUS, utilises the sensitivity matrices to iteratively calculate improved optimal values of the design parameters b, subject to the constraints imposed. The Pshenichny-Lim­Belegundu-Arora recursive quadratic programming (PLBA RQP) method is employed for this purpose.

·Although computational structural optimisation is still in its infancy, innovation in computer technology and recent developments in computational mechanics are rapidly enabling the analysis and structural optimisation of complex large-scale ap­plications.

Mechanics, DSA and Structural Optimisation

Research focused on the optimisation of linear structures, as a specific field of interest in engineering mechanics, span almost thirty years. The majority of linear optimi­sation problems could be solved purely analytically or with relatively simple com­putational procedures. For the analysis of applications with nonlinearity, however, the computational approach is vastly more efficient. With the introduction of high­speed, affordable computing power in the early 1970's numerical optimisation of complex structures with nonlinear behavior started becoming increasingly viable.

From the start of the 1980's researchers (15] took advantage of the advances made in computational structural analysis and proceeded to implement structural optimisa­tion in conjunction with various discrete (numeri~al) procedures, e.g. Rayleigh-Ritz, Galerkin, the FEM and the BEM. The isoparametric formulation (FEM) is espe­cially convenient for the purposes of performing DSA as the interval of integration is independent of the design variable (14], i.e. even for shape DSA the parent element formulation is independent of the design para~eters. Yang and Botkin(40] demon­strated the equivalence of the FEM and variational description of DSA for linear problems; this equivalence can also be demonstrated for nonlinear problems (35].

The initial applications of the FEM to structural optimisation were somewhat uri­realistic: only non-shape DSA was utilised and the kinematic and material mo­dels were mostly linear with no provision being made for more sophisticated path­dependent material behaviour. Later the investigation of DSA and structural opti­misation in conjunction with FEM methodologies capable of dealing with full ma­terial and kinematic nonlinearity started gainingground. Thus, in contrast to the wealth of literature available for struct;ural optimisation of linearly behaved struc­"tures, research in the domain of nonlinearity - particularly for material nonlinearity - is relatively young and not characterised by a lack of controversy.

The increasing application of DSA and structural optimisation to a wide variety of large scale real-world problems necessitated the development of optimisation models

3

capable of more accurate behavior. Much of this development is focused on more complex nonlinear material models which include path-dependent, and often also rate-dependent, behavior. Additionally, a substantial portion of the literature now exclusively focuses on large deformation scenarios, i.e. kinematic nonlinearity with dynamic and static constraints.

Due to the sheer complexity of issues relating to kinematic and material nonlinearity, sensitivity analysis and optimisation, research on the subject has been somewhat incremental. Ohsaki[28] presents a partial review of contemporary developments, particularly in regard to material nonlinearity:

• DSA for nonlinear elasticity was extended to idealised materials with perfectly plastic behavior; Kaneko and Maier[21) discussed the optimum design problem of plastic structures with strain hardening where only monotonic loading was considered.

• Path-independent methods of elasto-plasticity, using only monotonic loading, were investigated by Bends0e and Sokolowski(6]; no allowance was made for geometric nonlinearity.

• Employing a relatively straightforward piecewise linear constitutive model, Ray et al.[30] pointed out that the yield time of elastic-plastic constitutive models depends on the design parameters.

• Tsay and Arora(35] developed a method which successfully dealt with both path-dependent elastic-plastic material behavior and geometric nonlinearity, with good results.

Considering the effect of the path dependent solution process on DSA is vital in enabling the implementation of materially nonlinear optimisation models. Originally this issue was side-stepped by using nonlinear elasticity or elasto-plasticity without cyclic loading. 2 As a result the DSA formulation for problems with path dependent response is also path dependent. For path independent problems the DSA of the final configuration does not require sensitivity coefficients at any other increment during the load cycle (35][28)(36].

A further complexity resulting from the DSA of path-dependent nonlinearity involves the treatment of hardening and the attendant radial-return solution procedures. Kaneko and Maier(21) dealt with the optimal design for linear hardening or non­hardening plastic structures under proportional loading where constitutive law is reversible and path independent. Application to distributed parameter structures with kinematic and/ or isotropic hardening using the Von Mises yield condition and

2When the material goes into the inelastic range and unloading and reloading of the structure occurs (cyclic loading), the response variables are loading path dependent, i.e. the state of stress in the body depends on the full history of the state variables.

4

the elastic predictor radial-return method is discussed in (28]. Although isotropic hardening behavior is amenable to DSA, much controversy surrounds the application of DSA to material models involving kinematic hardening.

Rate-dependent material models add substantial complication to the implementa­tion of DSA. The expansiveness of this subject field means that a fixed body of theory for DSA applied to rate-dependent models is not likely to evolve soon. Yet a couple of standard approaches have been successfully applied to rate-dependent inelasticity, viscoelasticity and viscoplasticity. The preferred methods are based on implicit integration schemes. Additionally, some authors have enlisted novel mater­ial models in order to simplify the sensitivity analysis process, e.g. the endochronic constitutive model[20].

Performing a non-linear dynamic analysis as part of an automated design procedure, for a material with rate-dependent response, requires efficient techniques for calcu­lating the sensitivity of the non-linear dynamic response to variations in the design variables (24]. For structural analysis, explicit (forward Euler) solution procedures are usually faster than the implicit approach, while definitely lagging in accuracy. In determining the design sensitivity coefficients (DSC's) for applications involving both material and geometrical non-linearities, however, implicit solution procedures are more accurate and efficient [25]. This is a consequence of the fact that for DSA the explicit procedure has to use a small step-size to maintain accuracy and stability.

The implicit solution procedure is based on objective schemes that employ the Jau­mann and Green-Naghdi stress rates so as to ensure accurate integration of the con­stitutive laws. The implicit method, with objective integration, has been applied to a wide range of problems, including hypoelastic and viscoplastic materials [25]. Vidal et al.(37] and Vidal and Haber(36] used implicit schemes to obtain sensitivity coefficients for creep and elasto-plastic materials.

In working with material models with path and time dependency the issue of con­sistent linearisation and its implication for DSA has been voiced by several authors. Vidal[37] mentioned the consistent tangent operator as a requirement for the accu­rate implementation of creep in order to achieve enhanced accuracy. It has been demonstrated that, should direct differentiation DSA be performed, then an accu­rate formulation of the DSA procedure demands the use of the consistent tangent modulus, regardless of whether the analysis model uses it or not.

Due to the computational costs attendant in implementing a full structural optimi­sation procedure, especially for complex material nonlinearity cases, the majority of structures considered in literature are composed of one dimensional truss or bar elements. Some work has, however, been done with higher order elements:

• Rodrigues et al.[29) investigated the optimal design of a 2-D linear-elastic solid subject to thermal loads.

5

• Kulkarni et al.[24] performed material and viscoplastic parameter DSA on the viscoplastic response of thin plate structures subject to in-plane dynamic loading.

Although a wide range of sophisticated DSA and structural optimisation software has been developed in recent years, most of these codes are purpose-built. That is, very specific application orientated preprocessor and solver routines were de­signed and implemented to support the DSA and optimisation routines. Several authors have investigated the incorporation of sensitivity analysis and structural optimisation into established, usually FEM-based, commercial analysis codes. The implementation of design sensitivity analysis into the FEA code ADINA was in­vestigated by Haririan et al.[14] and Arora et al.[1], amongst others: as is the case with ABAQUS, structural optimisation cannot be included directly into the analysis package - the optimisation module operates in conjunction with the analysis module, i.e. the ADINA solver. The advantages of this approach are obvious: specialised features and facilities included with the analysis codes, e.g. contact procedures, wavefront optimisation and solver routines, do not have to be developed.

Nonlinear kinematic models and sophisticated material behavior, accounting for path and rate dependence, subject to cyclic loading and hardening represents the current state of the art in DSA and structural optimisation literature:

• A DSA method for structures with non-linear piecewise continuously differenti­able constitutive equations subject to cyclic loading conditions was presented in [28) by Ohsaki and Arora. Yield time is considered to be a function of design variables. Application to distributed parameter structures with kin­ematic and/or isotropic hardening is discussed using the Von Mises yield con­dition and the elastic-predictor radial-return method; geometric nonlinearity is assumed. Large variations in the sensitivity results, and therefore optimal values, result when loads are increased and yield occurs first at different points. Geometric nonlinearity was shown to have a negligible influence. Addition­ally, path-dependent material behavior causes DSA to be path-dependent, i.e. the DSA model is required to incorporate sensitivity estimates from all the previous time steps.

• Vidal(36) extended previous research done on the sensitivity analysis of systems with creep behavior and time-independent elastoplasticity. The treatment of algorithmic constitutive models is based on implicit integration strategies and return mappings associated with the elastoplastic consistency conditions. Consistent linearisation ensured that the system matrix appearing in the in­cremental sensitivity equations is identical to the consistent tangent stiffness matrix from the final equilibrium iteration of the time step.

6

Fundamentals of DSA and Structural Optimisation

Design Sensitivity Analysis

The last 20 years of active research into sensitivity analysis and structural optimi­sation has produced two primary analytical sensitivity analysis methods to the fore, namely the DDM (direct differentiation method) and the AVM/ ASM (adjoint vari­able/system method). Various other methods, e.g. the control volume and material derivative approaches, have been developed but do not appear to have gained the popularity of the ASM or DDM, particularly when used in conjunction with the FEM.

For exceedingly simple structural optimisation problems the FDM (finite difference method) may also be attempted: the FDM is a numerical DSA procedure and estimates structural response sensitivities by systematically performing perturba­tions of the design parameters and observing the changes in the response variables. This method is extremely inefficient when applied to realistic large-scale engineer­ing systems, in addition to becoming significantly inaccurate for a whole range of nonlinear applications. The FDM does, however, fulfill a useful role in enabling a relative straightforward verification of the actual analytical model.

The DDM involves taking variations of the continuum equilibrium equation with regard to the design variables. Often the stiffnes~ matrix of the analysis phase can be used as part of the DDM DSA procedure·, thereby greatly increasing the efficiency of the sensitivity analysis phase; this technique is here referred to as the 'intuitive' approach. It has been pointed out in the literature that should the material model require consistent linearisation, then the DDM procedure should utilise the consist­ent tangent modulus. As mentioned by Kleiber [22], one of the characteristics of the DDM is that the DSA expression is linear in displacement sensitivity and no incremental solution procedure is therefore necessary. This holds true for the ASM as well.

The ASM defines adjoint structures whose solutions permit explicit evaluation of the desired sensitivities. Adjoint sensitivity methods have been derived from virtual work methods, techniques based on reciprocal theorems, mutual energy methods and Lagrangian (weighted residual) methods [37]. In practice the ASM is only of benefit in path-independent problems for which the number of design variables is greater than the number of constraints. Therefore the choice between the two approaches depends on the ratio of active constraints and design variables, as well as the relative difficulty in obtaining the adjoint solutions vs. the sensitivity solutions [35]. The ASM is competitive against the DDM for path-independent problems only [22].

The main differences between the ASM and DDM are:

7

• The DDM is conceptually simpler, and easier to implement, than the ASM. In most instances it is also more efficient.

• The DDM automatically generates sensitivity data for the response fields at every point in the continuum; the information thus obtained enables the gen­eration of sensitivities for any number of performance functionals. In contrast the AVM only generates sensitivity information for an a priori specified set of performance functionals [37]. This difference is crucial for path-dependent problems since the history of calculated sensitivity vectors for every load level needs to be kept in order to calculate the sensitivity at a specific load level -this makes the ASM very tedious to use [35]. Adjoint methods are generally not very well suited for forward-marching, step-by-step methods.

Some work has been done towards formally establishing the theoretical premises for the discretisation of the general DDM and ASM variational formulations [2]. Sys­tematic proofs have been delivered to demonstrate the correctness of the numerical DSA formulations for linear and nonlinear problems.

Calculating the response sensitivities for transient dynamic problems can be shown to be equivalent to the process undertaken in obtaining the sensitivities for path dependent problems - the time history of the sensitivities of the state variables are required in order to calculate sensitivities at a further time t. Ray et al.[30] calcu­lated the derivatives of transient dynamic response variables as a special problem.

Certain optimisation procedures can take advantage of second order sensitivity in­formation so as to accelerate the optimisation process. To this end variational approaches which calculate the first and second order sensitivities of nonlinear sys­tems were investigated in various publications, notably by Mroz, Dems, Haftka and Haug.

Design sensitivity analysis procedures can be categorised depending on the design parameter with regard to which the response variable sensitivities are obtained: shape design parameters refer to variables which affect or determine the design of a structure, e.g. the base area of a pylon, whereas topological design parameters refer to the number of nodes, the types of elements, etc. Material parameters, e.g. Young's modulus and the Poisson ratio, belong to the category of non-shape design parameters. Although the conceptual difference is straightforward to assimilate, the sensitivity analysis procedures for the various categories are radically different.

A popular, 'unified' shape and non-shape DSA formulation based on the mutual Hu-Washizu principle and an Euler-Lagrangian kinematic description was presen­ted by Haber[16]. The 'unified' approach depends crucially on the reference volume concept, thus requiring a Total Lagrangian analysis formulation. This method en­ables a straightforward integration of the shape DSA procedure into a commercial FEM code. For modeling design optimisation problems the concept of the refer-

8

ence volume is shown to be equivalent to, and therefore compatible with, the FEM notion of a parent element that is invariant subject to changes in design [8]: the parent element used in the isoparametric mapping can be considered a reference volume. Alternative procedures, eg. domain parametrization and the material de­rivative approach, have been employed by some authors to perform shape sensitivity analysis.

Topological sensitivity analysis is substantially more complex than shape or non­shape DSA and has only recently been rendered practical with the application of the homogenisation method by Bends¢e and Kikuchi [7]. They proposed a generalisation of the topology DSA problem via the introduction of a material distribution model based on a porous medium with periodic microstructure: at every point in space the topological DSA procedure determines the presence (or not) of material. This model allows for the variation in the number and type of the elements used to model the microstructure. The material (homogenised) properties - elastic and thermal expansion coefficients - are computed using the homogenisation method [29]. The model has application to plate and shell reinforcement, natural frequencies control and multiload problems, to mention but a few.

Utilising the basic methodologies described above, variant forms of sensitivity ana­lysis have been applied to problems diversely involving elasto-statics and elasto­dynamics, eigenvalue and natural frequency analysis of beams and plates [37](12], thermal and thermo-mechanical effects, path-dependence, time~dependence, buck­ling, etc. Generally authors report sizable differences between sensitivity results with and without nonlinear effects.

In addition to applications focused purely on structural optimisation, design sen­sitivity analysis also plays an important role in a range of other engineering ap­plications, some of which are very rapidly gaining prominence. These applications include reliability analysis, inverse problems and parameter identification problems:

• Sensitivity information gathered from the DSA model may be used to assess the effect of uncertainties in the mathematical model and to predict response variations due to experimental inaccuracies in the design parameters. Re­search into these aspects extends the sensitivity analysis procedure via the introduction of probability theory and stochastic models; this forms the basis for reliability analysis. Deterministic and stochastic DSA for static and dy­namic problems were investigated by Kleiber, Hien and others [18)(19)(24][17] employing displacement-based FE modelling and a stochastic Hamilton's vari­ational principle.

For deterministic systems the gradient of the response functional with respect to the design variables is evaluated, whereas the subject of stochastic struc­tural sensitivity analysis is the estimation of probabilistic distributions of this gradient. As a result of the research in [17] a FE code was developed for the

9

deterministic and stochastic analysis of statics; minimum-weight optimisation; buckling load, eigenpair, forced-vibration and eigenvalue sensitivity analysis; obtaining static and dynamic response sensitivities for truss, beam or shell structures.

• Inverse problems [27] are concerned with the identification of structural parame­ters. The structural parameters are identified by presetting response variables on the structure boundaries in accordance with data obtained from experi­mental observations. The objective functional to be minimised is then for­mulated to be the difference between predicted and measured values of state fields.

Structural Optimisation

The aim of the standard structural optimisation procedures is to optimise a perform­ance functional subject to constraints on stresses and displacements; the structure may be subjected to static or dynamic, monotonic or cyclic loads. Applications include diverse problems in mechanical systems design, kinematic synthesis, shape determination and optimal control systems [34]. Rather than having to labouri­ously alter and reanalyse structural designs until the structural response matches the design constraints, structural optimisation seeks to automate the process.

Satisfactory structural optimisation crucially depends on four issues [9]:

1. The proper formulation of the optimisation problem.

11. The robustness, efficiency and accuracy of the optimisation algorithm.

m. The realism of the structural model that is used.

IV. Efficient structural and sensitivity analysis.

Mathematical programming has been widely adopted as the primary vehicle for structural optimisation [4][9]. Much work has been done in developing design optimi­sation models based on the standard mathematical programming (MP) techniques. Most of the problems in the MP literature, however, are formulated in terms of explicit functions, are of small size, do not possess a large number of local minima and do not require much effort for function and gradient evaluation, unlike optimal structure design applications. In consequence an entire field of optimisation research has come into being, focused on optimisation problems specific to structural design. Design sensitivity analysis is one of the most important bodies of theory developed specifically for the purposes of performing structural optimisation.

10

Important features in the study of optimal structure design problems are [4]:

• The problems are general - no restriction is placed on the form of the cost and constraint functions.

• The cost and constraint functions usually have an implicit dependence on the design variables.

• Cost and constraint functions evaluation is usually expensive.

• The optimal design problem is usually highly nonlinear and, in general, also non-convex. As a result it can have multiple local minima; globally convergent optimisation procedure are critically important in enabling structural optimi­sation.

• The problems are typically large; n=50 cost functionals and m=500 constraints is not uncommon.

The iterative approach to structural optimisation has matured since the 1960's and currently considerable research effort is being focused on the optimisation of non­linear structures - structural optimisation based on nonlinear formulations produces substantially different results to models which employ linearity. Integrating struc­tural optimisation with commercial analysis packages, like ADINA [14], is an addi­tional goal in a significant portion of the publications.

Research has shown that the optimal calculation depends heavily on the number of load increments. Load increment size has a considerable influence on the accuracy of the sensitivity coefficients - the DSC's are on average more sensitive to the size of the load increments than the response variables. Response variables that are good enough for convergence of the analysis procedure may not be accurate enough to serve as sensitivity coefficients.

The methods used in structural optimisation can be broadly categorised into (4)

1. Primal methods, which include recursive quadratic programming (RQP), gradi­ent projection, the reduced gradient method, the method of Bard and Green­stadt, the feasible directions method, an optimality criteria method, sequential linear programming and projection methods.

2. Transformation methods, which include sequential unconstrained minimisa­tion techniques (penalty and barrier functions) and multiplier (or augmented Lagrangian) methods.

An extensive comparative study of nonlinear programming methods for structural optimisation was presented by Belegundu and Arora(32). In (32] multiplier (i.e.

11

transformation) methods were argued to be less efficient than direct linearisation (i.e. primal) methods. Enhanced multiplier procedures, like the trust region method, have been introduced to increase the efficiency of the unconstrained optimisation process.

The efficiency and robustness of the quadratic programming methods contribute to the general appeal of the currently popular SQP /RQP (successive/recursive quad­ratic programming) procedures in the structural optimisation literature. The ef­ficiency and robustness of the SQP /RQP methods hinges on the superlinear and global convergence characteristics of some of the formulations available. 3 The SQP methods are based on a theorem which states that the optimum is the global min­imum of a Lagrange function in the subspace of vectors orthogonal to the active constraint gradients or the tangent subspace. The Hessian of the Lagrange func­tion is not calculated explicitly, rather an algorithmic approximation procedure is constructed to update the Hessian of the Lagrange function.

The SQP methods all have the following qualities [34):

• Global convergence.

• Superlinear convergence of the iterative process near the solution point if step length is unity.

• Use of only first order information.

• Natural suitability to equality and inequality constraints.

Differences which exist between the various SQP techniques are generally a con­sequence of variations in the definition of the QP subproblem solved at each iter­ation and the Lagrange descent function used during step-size determination. The performance of the optimisation algorithm can be radically influenced depending on the appropriateness of the SQP method used. Infeasibilty of the QP subproblem and the loss of the local superlinear rate of convergence are some of the major problems faced with the SQP problems.

A particularly efficient and robust method is the PLBA (Pshenichny-Lim-Belegundu­Arora) RQP. The PLBA is based on Pshenichny's descent function, preferred to other standard descent functions like the augmented Lagrangian function used by Schittkowski or a LI penalty function. The PLBA has the robustness qualities of the Pshenichny method, with increased efficiency: the PLBA is globally convergent,

3 The SQP /RQP superlinearly convergent solution techniques, employed to solve nonlinear pro­gramming (NLP) problems, belong to the class of procedures variously referred to as superlinearly convergent recursive quadratic programming (RQP), projected Lagrangian, successive quadratic programming (SQP) or constrained variable metric (CVM) methods [34]. The SQP methods in general require fewer function and gradient evaluations than the standard quadratic programming algorithms.

12

uses an active set strategy and superlinearity is achieved via a BFGS acceleration of the Hessian of the Lagrange functional.

Thesis Focus

-As mentioned, the main aims of this thesis are:

• Obtaining a DSA formulation which could be discretised and, in theory, be implemented as a 'pseudo'-element in any established FEA code.

• Implementing structural optimisation in conjunction with ABAQUS to vali­date the success of the 'pseudo' -element approach.

These aims were formulated so as to lay the foundation for further research into structural optimisation, specifically with ABAQUS.

In order to make the range of tractable optimisation problems4 as wide as possible, more emphasis has been placed on sensitivity analysis, as opposed to optimisation.

A large portion of the effort invested in this work has focussed on the integration of DSA into ABAQUS and implementing an optimisation model capable of interfa­cing with ABAQUS. As described in Chapter 5, the user element routine UEL was developed so as to allow alternate execution of structural and sensitivity analysis increments - the discretisation of the DSA formulation as a 'pseudo'-element ensures that the native ABAQUS solver interprets the DSA increment as a structural ana­lysis step. Though not as complex, implementing optimisation in conjunction with ABAQUS is more labourious, requiring the development of various script files and utility programs to manage the interface between ABAQUS and the optimisation model.

In order to curb computational cost and to maintain the focus .of this research, the nonlinear kinematic formulations employed are simplified: several assumptions are introduced and a linear elastic truss/bar element discretisation is used consistently.

Only shape (geometric) and non-shape DSA is investigated and implemented (cf. Section 3.1.1 ): possible response variable dependence on load and boundary con­ditions, as well as topological characteristics1', is. not considered. For the sake of convenience, the truss/bar element crqss-section is consistently used as a non-shape design parameter, while Fig.(6.4.3] defines the shape design parameter used in Sec­tion 6.4.

4 Shape and non-shape optimisation; multi-parameters DSA; stress-based DSA; etc. 5 Number of nodes, elements, connections.

13

In order to keep the structural optimisation procedure as simple as possible, and to enable compatability with ABAQUS, the DSA formulations utilised are all based on the Direct Differentiation Method (DDM). In brief, the three main DDM formula­tions are:

• The 'intuitive' non-shape approach which utilises the stiffness matrix K, ge­nerated during the analysis phase, to obtain approximate sensitivity estimates. For the purposes of this research the 'intuitive' approach incorporates only the Updated Lagrange {UL) kinematic formulation, although the Total Lagrange (TL) formulation is equally well suited.

• The 'rigorous' non-shape approach which, unlike the 'intuitive' approach, ap­plies the complete DDM analytical procedure to the kinematic formulation. In this work the TL formulation is preferred to the UL to ensure a more accurate large strain sensitivity response. Although more accurate than the 'intuitive' approach, the 'rigorous' procedure is substantially more complex, in theory and implementation.

• Shape DSA, employing the TL-based 'rigorous' approach. The infeasibility of UL-based shape sensitivity analysis follows from the formulation for shape DSA in Chapter 36 •

In conclusion, the PLBA RQP optimisation procedure was implemented to work in tandem with ABAQUS to obtain the optimal structural design configuration. Due to the emphasis of this research on DSA and the refinement of the PLBA, the optimisation routine is utilised as presented in [26], with minimum modification.

Sectional Outline

The rest of this thesis is constituted as follows:

• Chapter 2 is concerned with a review of fundamental kinematic theory, an introduction to the terminology and notation used in the ensuing chapters and a discussion of the nonlinear kinematic solution procedures. The discretisation of the various kinematic formulations is also provided.

• Chapter 3 introduces the concepts and theory employed in the formulation of Design Sensitivity Analysis: the variational formulations for the 'intuitive' and 'rigorous' non-shape DSA approaches are presented; the same is done for shape and stress-based DSA. Once again the variational expressions are discretised.

6 Refer to Fig.[3.1].

\

14

• Chapter 4 discusses and summarises the structural optimisation procedure implemented in this research, namely the Pshenichny-Lim-Belegundu-Arora recursive quadratic programming (PLBA RQP) method.

• Chapter 5 describes the integration of the kinematic and DSA formulations into the ABAQUS user element subroutine UEL. In addition, the structural optimisation process, implemented in conjunction with ABAQUS, is detailed.

• Chapter 6 presents numerical results for several structures, demonstrating the successful application of the theory introduced in Chapters 2 to 4.

• Chapter 7 concludes the thesis.

Chapter 2

Structural Analysis

2.1 Variational Formulation

The computational implementation of Design Sensitivity Analysis (DSA), and con­sequently most numerical Structural Optimisation procedures, requires extensive analytical manipulation of the classical equation of motion1 in variational form. In the ensuing chapters and sections, familiarity with the standard variational formu­lations and notations will be assumed. For the sake of clarity, a brief review of the fundamentals of the standard displacement based variational formulation, as well as kinematic nonlinearity, will be presented.

2.Ll First Principles: The displacement-based equilibrium formulation

The increasing availability of affordable computational power has not only enabled more complex structural analyses to be performed, but has also made computational design sensitivity analysis, and thus structural optimisation, more feasible for a wide range of applications. Substantial research effort has, in addition, been focussed on producing variational balance laws which result in less expensive computational implementations.

From this perspective the specific variational description employed to model the equilibrium state is crucial in maximising computational efficiency, especially in the case of material or kinematic nonlinearity. Although promising alternative element formulations exist, eg. the three-field 2 (Hu-Washizu) based enhanced assumed strain

1 Also: governing or equilibrium equation 2Three variational equilibrium equations are solved simultaneously for three different response

15

16

approach, the standard quasi-static one-field displacement-based formulation is ad­opted throughout this work due to its relative simplicity, particularly for nonlinear behaviour, albeit with some loss of computational efficiency.

For a static or quasi-static3 body, the linear momentum equilibrium equation[ 33J

simplifies to:

;:Jr-· U I) f -+p ·=pa· Bxi 3 3

01'." -2.. +pf-= 0 OXj J

(or V • T + f = 0 in vector notation).

(2.1)

The traction boundary condition on the surface4 • , S, of a static or quasi-static body, is generally expressed in the following manner: .

(2.2)

(or r·n - t = 0 in vector notation).

Displacement BC Traction BC

r;.-~--t--- / i;

St Venant's Principle

Figure 2.1: Static or Quasi-static body with Boundary Conditions

Applying the principle of virtual work5• to (2.1) and (2.2), and combining them, the

standard displacement-based variational formulation is obtained. The variational form of the governing equation, in conjunction with the essential boundary condition; ie. prescribed displacement U On r Ul fully describes the problem.

variables [39]. 3Incremental loading is applied in a fashion such that linear acceleration remains negligibly

small. 4 Scan be divided into the surface of prescribed displacement r u (Essential BC), and the surface

of prescribed traction rt (Natural BC) 5 A variation on the general Lagrange Multiplier method [39], as opposed to taking the

Frechet/Gateaux derivative of the Total Potential Energy [9]

The virtual work form of expression (2.1) is

[ (orii + pfi) bui dV = 0 lv oxi

=> [ OTij bui dV = - f Pfi bui dV lv axj lv

and applying a similar procedure to (2.2), the following relation is obtained:

ls (rijnj-ti) oui dS = 0

a => f -

8 ( rijoui) dV = f ti bui dS lv Xj ls

17

(2.3)

=> [ arij OUj dV + [ Tij OOUj dV = [ ti bui dS. (2.4) lv axj lv OXj ls

Consequently, combining (2.3) and (2.4) produces

1 r·. aoui dV -1 pr. ou· dV = 1 t· bu· dS iJ a J, , , ·'

V Xj V S

i.e. iv Tij oeij dV - iv pfi bui dV =ls ti bui dS (2.5)

and, more specifically, for a displacement based formulation

f ti ouidS = f ti bui dft ls lr1 or, using vector notation:

f TT •Oe dV - f pf T ·De dV = f t T bu df t, lv lv lr1

where the Cauchy6 stress vector is

TT = [ Tu T22 733 T12 T13 T23 ] '

and the infinitesimal strain vector is

e T = [ eu e22 e33 /12 /13 /23 ] ·

(2.6)

For the infinitesimal displacement domain the formulation (2.6) is quite satisfactory. When applied for the purposes of solving problems involving large deformation, however, especially when accompanied by large strain response, the infinitesimal strain formulation (2.6) expressed in terms of Cauchy stress ( Tij) and infinitesinal strain ( eij) is no longer adequate.

Most of the research performed here ·focuses on finite displacement applications, therefore requiring a kinematically nonlinear formulation. Two nonlinear formula­tions capable of correctly modelling large deformation behaviour, specifically the Total (TL) and Updated LagrangiaIJ. (UL) procedures, are discussed in the next section.

6Utilising Cauchy stress is justified by assuming a static or quasi-static body.

18

2.1.2 Kinematic Nonlinearity

Assuming equilibrium at time t, (2.5) is restated more conveniently as

for a small rotation and displacement problem. The variational formulation for a further time t + tlt, ie. for altered loading conditions t+At~, is expressed as:

(2.7)

with the external loads t+At~ formulated as:

t+At~ = J t+Atp t+Atfi bt+AtUi t+AtdV t+~tV

+ J t+Atti bt+AtUi t+Atdft. t+6tft

In the event of large rotations and/or displacements, that is, kinematic nonlinearity, the equilibrium state at time t +tit is formulated in terms of the material reference state at time t = 0, employing the Total Lagrangian (TL) procedure, or the material configuration at time t, utilising the Updated Lagrangian (UL)7 approach.

The strain energy measure Tij 8eij, in the small displacement/rotation variational formulation, is invariant to frame transformation and can therefore not be used to adequately model the equilibrium state at time t +tit in terms of either of the frames Co or Ct.

The energetically conjugate8 , frame invariant? 2nd Piola-Kirchhoff stress Sii and

Green-Lagrange strain €ij tensors respectively replace Cauchy stress and infinites­imal strain in both the TL and UL formulations.

In the work presented here, both the TL and UL are employed and consequently both formulations will be presented. The notational conventions introduced by Bathe [3] are used consistently in the ensuing sections. ·

7 Refer to Figs.(2.2-2.3].

8 Energy conjugacy ::} J t+At Sij 15t+At C:ij t+At dV = J t+At r;; 15t+At e;; t+At dV , •+a•v •+a•v

9 Frame invariance implies that for any two frames T( 1) and TC2 ), where either frame can be

obtained by a frame transformation of the other, the following relation holds true

J r<'' S;; 15r<'' c:;i r<'> dV = j T(21 S;; 15r<2> c:;; r<2> dV r<'! V r(2) V .

19

The Total Lagrange kinematically nonlinear formulation

-r---------------~xo

Figure 2.2: The Total Lagrangian (TL) formulation.

In the Total Lagrangian 10 variational formulation, the equilibrium state (2. 7) at time t + D.t is expressed in terms of quantities defined in the 'initial' frame11 C

0

(2.8)

In order to determine the response variables, eg. stress, at time t + D.t, (2.9) is solved for the incremental displacement t+t.tUj whence ~+t.tUj = ~Uj + oUi is obtained. The displacement ~ui is found from the solution of the equilibrium state at time t

10Refer to Fig.[2.2]. 11 Note:

• '[]'indicates a 2nd Order Tensor.

0 • As a consequence of Conservation of Mass[33], det ([ ~+Atxk,1]) = -.:iirp.

• Assuming an incompressible material or rigid body displacement, det ([ ~+Atxk,i]) = 1.

20

Assuming that the body and traction forces are not 'follower forces' 12 , t+ilt~ is utilised in the form (2.9), for both TL and UL.

In order to solve the nonlinear TL equation for the displacement increments oui, (2.8) can be written as

f oCiirs oers 8oeii 0 dV + j ~Sii 8o'r/ii

0 dV ov ov

~ t+ilt3? - j ~Sij 8oeij 0 dV ov (2.10)

using the set of simplifi.cati~ns and linearisations introduced below13 •

• Displacement and strain increments:

The displacement and strain increments in TL refer to the 'initial' refer­ence frame

where

• Simplifications:

t+iltu. 0 I

t+ilt c .. Q '-'IJ

t 0ui + oUi

~cij + o£ij

t l(t t + t t ) QCij = 2 oUi,j + oUj,i oUk,i oUk,j .

The incremental form of the standard response variables at t + .b.t is14

where o£ij = oeij + OT/ii

and oeii = H oUi,j + oUj,i)

l(t + t + ) OT/ij = 2 oUk,i oUk,j oUk,j oUk,i oUk,i oUk,j

for which the variational formulation is

with 8oeii =

8ortii = H 8o'IJ,i,j + 8ouj,i) 1 t ~ t 2( 0 uk,i 8ouk,j + 0 uk,j 8ouk,i

+ bouk,i ouk,j + ouk,i bouk,j) .

(2.11).

(2.12) 12 1 +~ 1 lR in (2.9) is subject to the assumption that the body fi and traction forces ti are not

dependent on the process of deformation, ie. they have no dependency on ~+~tui. 13All variations are taken with respect to oui: 8 = 8

0u;·

14The incremental form of the response variables in some instances differs radically for the UL

for 111 ulation.

21

A further simplification is usually introduced by eliminating some of the . 15 s: terms m uoTJij

In order to achieve a higher degree of accuracy in the TL formulation, DoT/ij is used as introduced in (2.12) in the research presented here.

• Linearisations:

The nonlinear term of (2.8), ie. containing the incremental 2nd Piola­Kirchhoff stress oSij, is linearised by replacing the variation of the Green­Lagrange strain 0€;j with the variation of infinitesimal strain Oeij·

oSij is itself linearised17 :

~+b.t S;j ~Sij + oSij = ~Sii + oCijrs oEij

~ ~Sij + oCiirs oers .

In addition to calculating the response variables (stress, strain), the displacement re­sponse ~+b.tui also plays a critical role during the sensitivity analysis phase, whence the sensitivity vectors are obtained to eventually be used in the structural optimi­sation procedure. The numerical procedure employed to solve for ~+b.tui is based on the straightforward Newton-Raphson approach:

• The linearised TL formulation represents a 'predictor' stage in the process of determining ~+b.tu;. Errors occur in the estimate of ~+b.tui due to the linearisations introduced into the formulation of the TL equilibrium equation.

• An iterative Newton-Raphson (N-R) 'corrector' method is employed by the solver to calculate a succession of better estimates for ~+b.tui[3]

15Thus ensuring that the stiffness matrix K produced during discretisation remains symmetric

in the kinematically nonlinear regime, thereby reducing computational cost. 17 Assuming linear elasticity

22

which in the canonical vector form is simply

J C J: T A (k) odV + J. J: Ao'" T(k) • ot S odV o uoe · uoe uu ., oy oy

t+At~ - J 6t+Ate T(k-1) • t+At 5(k-1) odv. oy O O (2.13)

It has to be pointed out at this stage that the entire preceding analysis procedure pre-supposes an incompressible material or a material with very high values of Young's modulus (ie. essentially a rigid body), where effectively i+Atp = ip = 0 p.

The Updated Lagrange kinematically nonlinear formulation

y

~·· lt~IMtl

--4------x

Figure 2.3: The Updated Lagrangian (UL) formulation.

The TL formulation is preferred to the UL18 , as the former can be integrated more naturally into the ABAQUS user element routine. The TL is, furthermore, more ac­curate than the UL for large strain behaviour, as well as being more straightforward to implement for geometric (shape) design sensitivity analysis. Notwithstanding these facts, the UL is nevertheless extensively utilised due to its relatively simple formulation, as well as for comparative purposes.

Equilibrium for the UL formulation is succinctly expressed as

(2.14) 18Refer to Fig.[2.3].

23

with all expressions, except t+At~, referring to the 'current' (time t) material configu­ration frame. The term t+At~ remains unchanged from (2.9), assuming deformation independent loading.

The linearisation of the UL formulation in (2.14) is obtained as

(2.15)

derived by imposing a series of simplifications and linearisations, as discussed below. The process of reduction of the nonlinear equilibrium expression (2.14) to (2.15) proceeds in a similar manner to what was done in the case of the TL formulation:

• Displacement and strain increments:

The displacement and strain increments in UL refer to the 'current' ma­terial configuration frame at time t, ie. the last time increment for which a convergent solution was obtained ...

t+Atu· t t -

t+Atc:- .. t c..,tJ -

where

• Simplifications:

The incremental form of the deformation measures at t + lit is

tCij = teij + tT/ij ,

where teii = t{ tUi,j + tUj,i).

tT/ij = ~( tUk,i tUk,j) •

the variations of which may be stated as

with 8teii = !( 8tui,j + 8tUj,i)

8tT/ij = !( 8tUk,i tUk,j + tUk,i 8tUk,j)_

24

• Linearisations:

The LHS term of (2.14) containing the incremental stress tSij is linearised by approximating btcij ~ bteii so that

A further linearisation is imposed by assuming linear elasticity in the incremental stress t Sij

t+at "·. ts + s t + c t Ji3 t ij t ij = Tij t ijrs tCrs

Finally, in a process analogous to that adopted for the TL formulation, the iterative form of (2.15) is found to be

which in the canonical vector form becomes

(2.16)

25

2.2 Discretisation

Both the Total Lagrangian and Updated Lagrangian formulations are numerically solved by means of the Finite Element Method (FEM); the TL and UL procedures are both discretised in preparation for computational implementation as part of the FE analysis code ABAQUS' user element routine UEL.

The generality of the element discretisations employed in the work presented here . is limited by a series of assumptions, as well as the specific area of focus:

• The majority of physical problems to which structural optimisation is applied involve truss structures and, consequently, the bulk of the current literature in this subject field focuses on the FE discretisation and implementation of truss, or bar elements19

; consequently the research presented here is limited to the utilisation of bar elements exclusively.

• In the general variational formulation for the TL and UL the loading forces (traction and body forces) are assumed to be deformation independent, i.e. they are 'non-follower' forces.

• The material model is chosen to be linearly elastic.

• In a significant further assumption for the truss structures investigated, the self-weight of the trusses is considered negligible in comparison with the applied traction forces. Although this assumption does not model the majority of physical applications very realistically, it does simplify the DSA formulation without compromising the conceptual basis of the sensitivity calculation20 •

By eliminating body force in the variational formulation, t+At~ simplifies to

t+At c;o _ J t+Att. c' U. Odf ;n - 0 I UO I t •

0 rt

The discretised Total Lagrangian formulation

For a truss element implementation the iterative linearised TL equation (2.13) is restated, in the global 'initial' frame21 C0 , as:

J E Doexx ~Doe~~ 0dV + j 6~011i~ ~Sxx 0dV ov ov--19

ln order to avoid confusion with truss structures these elements are consistently referred to as bar elements.

20The body force term is treated similarly to the traction force term, assuming material incom- ,

pressibility, and its presence in the variational formulation therefore doesn't contribute significantly towards gaining further insight into the computational implementation of design sensitivity ana­lysis.

21 Refer to Fig. (2.4).

@ ~ +1

Natural/Parent Frame c,

translation

> © .,

11

LocaVPrlnclpal lnltlal Frame c' 0

rotation

>

Figure 2.4: Isoparametric mapping.

26

z

Global lnltlal Frame

The discretisation is substantially more straightforward to perform when the pre­ceding expression is formulated in the local 'initial' frame Co' , which is defined to be the princial frame of a particular element, ie. C0

1 is a frame in which both nodes of the element are mapped onto the local 0 :v1 axis: .

J (k) 01 J (k) t 01

E boiexx 6.801exx dV + Sf:l.01T/xx 0,Sxx dV ~v ~v .

- J bo1U T . ~tAtt O' drt - J So1ei~-l) ~tAt si~-l) 0' dV. o'r, o'v .. (2.17)

Implementing the FEM, discretisation proceeds in the usual fashion by first intro­ducing the standard isoparametric mapping from the 'natural' (or 'parent') frame Cr to the local/ principal 'initial' reference frame C0', so that (2.17) is restated as

J E boiexx 6.801ei~ J r dV + j' Sf:l.017Ji~ ~,Sxx J r dV rv . rv

- J bo1U T . ~tAtt Jr r drt - J So1ei~-l) ~tAt s~~-l} J r dV' (2.18) rr, rv .

where 'r' refers to the 'natural' configuration Cr, with the Jacobian coefficients J, Jr defined as

J - d et ( [0' x · ·] ) = 0

L r i,J . 2

Jr - II [~' Xi,inil II-' The isoparametric mapping Cr --. C0' is represented by a transformation operation

from a coordinate system in e to a system in °:-c', which can be written as

( ox') 0

:z:' = 0 :i:' ({) where 0:i:' = :~: ' e = ( n ' (2.19)

27

utilising the shape functions, N1 = ~ and N2 = 1¥, so that 0:c' is simply defined, in terms of the nodal coordinates X', as

. .

[ N1 0 0 N2 0 ;, ] 0 z' =NT· X' where T . 0 0 N2 N = 0 Ni

0 0 Ni 0 0

and X'T = [ X1 1 Yi' Z1 1 X2' Y2' Z2' ) . (2.20)

Hence, 0z' is a vector of first order continuous functions based on the interpolated

nodal coordinates X' of the truss element in the local 'initial' frame C0'. From 0z'

the orientation °z of the element in the global 'initial' frame C0 is obtained as

(2.21)

by applying the frame transformation

0TT = [~x;J] = [:::;,]

where the transformation °TT is expressed22 in terms of(} and </>as

oT T _ [ O~a O;a J with

[

cos </> cos (} - sin (} - sin </> cos (} ] T = cos </> sin (} cos (} - sin </> sin (} .

sin </> 0 cos </>

Referring to (2.20), X' is found by mapping the truss element nodal coordinates X from the global 'initial' frame into C0' in a procedure analogous to (2.21):

X'= 0T·X. (2.22)

Consequently, the localisation and isoparametric mapping processes in (2.20) and (2.22) may be combined as

(2.23)

As will become apparent in the section on shape DSA, the role fulfilled by the localising transformation matrix23

(2.24) "

in the discretised TL formulation is of considerable importance. The definition of T and its dependence on structural configuration are further examined in the relevant section.

22 Refer to Fig. (2.4). '.!

3h:: shape design parameters.

28

The isoparametric Cr~ C0 (with NT) and localising Co~ C0' (with 0T) mappings are employed to express the incremental displacements 0u' as interpolating functions of the global nodal incremental displacements d in the same manner that 0z' is expressed in terms of X in (2.23): ·

with

d'T -dT -

0u' NT· d'

NT. 0T·d

OU

[ U11 I I I I '] V1 W1 U2 V2 W2

[ U1 V1 W1 U2 V2 W2 ] (2.25)

where 0u and d are, respectively, the interpolated incremental displacement and nodal incremental displacement vectors in the global 'initial' frame C0 •

Finally, the FEM discretisation of (2.I 7) is completed by introducing (2.I9)-(2.25) into (2.I8):

Do1U T · ( { 0 A 0 L E 0 N x,x © 0 N x,x T} + { 0 A 0 L ~Sxx 0N}) · ~Do1U(k)

Do1u T. ( {jrr1 ON© ONT ~;Lltts r dI't }- { o Ao L oN(k-1) ~+LltSxx(k-1~}) (2.26)

where the symbols used in (2.26) are defined as: t c o'-'xx

OL ~1Sxx 0

' L = truss length oA - 0

' A = truss cross-section

E - Young's modulus

_I [-I OL 0 0 I 0 0 ]

OIL [ 0 -I 0 0 I 0 ]

OIL [ 0 0 -I 0 0 I ]

ON ONT xx© xx+ ' t

0 ~ 0 T Ny,x© N y,x+ ON ONT

. z,x © z,x

o N + oN . o'T . t+Lltu1(k-1) x,x 0

t+Cl.t 'i (k-1) o ~xx

t S + E [ o~ T (k-1) _ ! t T(k-1) . OTT . oT. u(k-1)] o xx -'"' 2 oU 0

(2.27)

29

and ~tAtt5 is a delta function describing the traction force distribution such that

with O'zi = O'zl(node=i) and

In preparation for implementation of the ABAQUS user element routine UEL, the discretised formulation (2.26) is expressed in the global 'initial' frame and simplified by expanding bS:1:x = E bcxx:

where

for

[ 0TT · o A o L ( { E ON x,x 0 o NT x,x} + { ~Sxx 0N}) · 0T] · ~oU(k)

[ i1x i1y i1z i2x i2y i2z ] ,

( o Ao L E) oN(k-1) . oN T (k-1)'

- (k-1) oN = o N x,x + ! oN . oT . ~+Atu(k-1) .

(2.28)

(2.29)

(2.30)

From (2.28) it is evident that the (k - l)th iterative estimate for stress b+AtSxx(k-l) becomes

-T t+Ats (k-1) = E oN (k-1) . oT. t+Atu(k-1) 0 xx - 0 (2.31)

The discretised equilibrium expression (2.28) is easily rewritten in the standard quasi Newton-Raphson iterative form

(2.32)

The discretised Updated Lagrangian formulation

The truss element implementation of the linearised UL equation (2.16) is expressed in the global 'current' frame Ct in a procedure similar to that used for the TL

30

formulation in the previous section:

J E Stexx b..Ste~~ tdV + J Sb..t1]~~ tTxx tdV 1v 1v

J Sou T. t+ti.tt odft - j St Atexx(k-1) t+AtT(k-1) t+Atdy(k-1). Oft 0 t+.dtV(k-1) + xx

In localising the variational expression (2.33), the UL formulation is dealt with in a significantly different manner to the TL by utilising separate mappings Ct -+ C't and Ct+6.t<k-1) -+ C't+At<k-1) for the LHS and RHS of (2.33), respectively. The isoparametric mappings C't -+ Cr and C't+At<A:-1) -+ Cr are then utilised to express the LHS and RHS of the localised expression (2.33) in the 'parent' frame.

J E Dt'exx b..St'e~~ Jr dV + J Sb..t'T/~~ t' Txx t' Jr dV rv rv

J Sou T • ~+Att Jr r df t - J St+6t'exx (k-1) t+At' Txx (k-1) t+6t' J(k-1) r dV. rrt rv

(2.33)

The Jacobian transformation factors are defined as follows:

t'J - det ([t'x· ·]) = .:__!: r 1,3 2

t+At' J(k-1) det ((;+At' x~~-l))) = t+At L(k-1)

2 Jr - II [~xi,ini] II ·

The isoparametric mappings Cr -+ C't and Cr -+ C't+At<k-1) are defined in the standard fashion:

• The incremental displacement in the local 'current' frame C't, 0u', is an in­terpolation based on the nodal incremental displacements td' in C't

tU1 =NT. td'

with

The incremental displacements in C' t are expressed in terms of the global nodal displacement vector t d as ·

tU' =NT. tT· td.

• In the RHS of (2.33) a similar procedure is utilised in order to define the inter­polation function for the incremental displacements t+Atu(k-l) in the (k - l)th iterative local 'current' frame C't+At<k-1)

t+fitU 1(k-1) NT . t+Atd1(k-l)

- NT . t+AtT(k-1) . t+Atd(k-1)

31

with

t+Atdi(k-l) = [ u' v' w' u' v' w' J . 1 . 1 1 2 2 2 t+At(k-1)

The defin~tion of the transformation matrices proceed in a straightforward fashion from the preceding section:

Transforming from C't to Ct

tT

with

and from C' t+6t to Ct+6t

with

- sin to cost()

0

- sin t</> cost() l - sin t</>sin to

cost</> .

[

cos ~ c~s ~ - sin 0 · cos </> sm 0 cos 0

sin J> 0

- sin~cos~] - sin </> sin ()

A

cos</>

h J, = t+At ..1.(k-l) (J" = t+At(J(k-1) w ere .,, _ .,, , _ .

The locally discretised form of (2.33) is

bt1U T · ( { tA tL E tNx,x ® tNx,x T} + { tA tL t'Txx tN }) · ~t'U(k) t+Att c UT(k-1) { t+AtA(k-1) t+AtL(k-1)} t+AtN (k-1) t+At..,.. (k-1) O - Ot+At' · x,x •xx ,

(2.34)

where the symbols used in (2.34) are defined as

t t' 1'xx - Txx

tA - t'A

tL t'L = t' X2 -

t' X1

t+At7

(k-1) t+At' T (k-1) xx - xx

t+AtA(k-t) t+At' A(k-1)

t+At L(k-1) t+At' L(k-1) = t+6t' X (k-1) 2 -

t+At' X (k-1) 1

E - Young's modulus

32

tNT _1 [ -1 0 0 1 0 0 ] x,x tL

tNT y,x - t~ [ 0 -1 0 0 1 0 ]

tNT z,x t~ [ 0 0 -1 0 0 1 J

t+AtNT (k-1) 1 [ -1 0 0 0 J x,x - t+At L(k-1) 0 1

t+Att T 0 - [ l1x t1y l1z l2x t2y l2z ]

tN = tN '°' tNT + x,x 161 x,x

Assuming material incompressibility t+Atp(k-1) tV

tp t+AtV(k-1)

tA tL --------- = 1, t+At A(k-1) t+At L(k-1)

the expression for t+AtTxx(k-l) in (2.34) simplifies as follows:

t+At7

(k-1) xx det [ t+Atx(k .. -1>] ( t+Atx(k-1)) ( t+At S (k-1)) ( t+Atx(k-1))

t t,J t 1,1 t xx t 1,1

(

t+At (k-1)) p ( t+AtX(k-1))

2 ( t+At S (k-1))

tp t 1,1 t xx

( t+AtX(k-1)) 2

( t+At S (k-1)) t 1,1 t xx

(<I>)2 (!+AtSxx(k-1)) (2.35)

where

<I> _ <I> (t</>, t+At</>(k-1)' to, t+AtO(k-1))

( t+At (k-1)) t X1,1

[ t+AtTT(k-1) . tT] ..

(1,3=1)

cos~ cost¢> (cos{) cos to +sin{) sin to)

The (k - l)th iterative stress estimate for the UL procedure takes on the form

:+At Sxx (k-l) tTxx + E ( t+AtN(k-l)) · t+AtU1(k-l)

for

tN = tNT + l u T • tTT. tN x,x 2t

Globalising the discretised UL formulation, (2.34) becomes

[ tTT · O AOL ( {Et N x,x ® t NT x,x} + { tTxx tN}) • tT] ·~tU(k)

iK ~+Att _ [ t+AtTT(k-1). _c(k-1). t+AtT(k-1). t+AtU(k-1)] (2.36)

33

where

_c(k-1) = {4>)2 { OA OL t+AtNx,x(k-1)} { tTxx + E ( t+AtN'k-1))} .

A comparison between {2.28) and {2.36) emphasises the substantial differences which exist between the TL and UL methodologies.

From (2.36) the standard quasi Newton-Raphson iterative form for the UL is ob­tained:

{2.37)

For the UL formulation the full Newton-Raphson iterative procedure could also have been employed, so that

t+AtK(k-1) A u(k) _ t+AtM _ t+AtQ(k-1) 'Ut - 0 :A. 0 '

Chapter 3

Design Sensitivjty Analysis

3.1 Variational Formulation

3.1.1 General Ideas on the formulation of Design Sensiti­vity Analysis

The object of DSA is to obtain a vector, \7 b '11 = ~'which expresses the sensitivity

of an arbitrary functional '11 = '11(b) subject to variation of the design parameters b.

For realistic applications the sensitivity matrix \7 b '11 is usually found for

'11 E {objective function, constraint functions} .

These sensitivities are crucial in implementing structural optimisation, for which DSA is commonly employed; that is, DSA has to solve for

i E le, c = {active constraints}

which represents the generalised cost functional J( u, b) and generalised constraints 9i( u, b ), respectively.

More specifically, for the research presented here,~the object of investigating DSA is to show that

1. Generalised DSA, specifically the direct differentiation method (DDM), can be reformulated and discretised in a manner analogous to the element formu­lations employed in Finite Element Analysis (FEA).

34

35

11. The FE analogous, or 'pseudo-element', formulation can be implemented in the ABAQUS user element routine UEL in order to perform sensitivity analysis, thus enabling the implementation of structural optimisation using ABAQUS.

For problems in engineering mechanics the design parameters b fall broadly into . four categories:1 [27]

1. Parameters which influence the elasticity (or elastic-predictor) matrix. ( ie. part of the material model)

2. Shape or configuration variables. (ie. the geometric design of a structure)

3. Support and loading conditions.

4. Topological support parameters, eg. number of joints, elements or connections.

The functional '11 may be quite general, but for applications in mechanics is usually formulated in terms of position :v, response variables - such as displacement u, strain e or stress T) - and design parameters b, e.g. (35]

The complicating factor in obtaining sensitivity vectors for problems in mechanics is the implicit dependence of '11 on the design parameters b

'11 = '11 ( ~+tltu, b) = '11 ( ~+tltu(b ), b) . Further complexity arises from the dependence of '11 on response variables other than displacement, e.g. t+tltT and t+tlte, especially where the response variables are implicitly formulated in terms of displacement, ~+tltu.

Using the chain rule, the sensitivity vector for. t~e functional '11 is simply obtained as

(3.1)

1 In this work only category 1 and 2 parameter sensitivities are investigated, generally referred

to as non-shape and shape sensitivity, respectively.

36

where :~, as well as the explicit dependency of Won b, ~~, can be derived analyt­ically in a straightforward fashion.

The main pursuit of computational DSA then, is to find the implicit dependency of u on b, i.e. ~. As described in the introduction, several methods exist which,

to varying degrees of success, enable the numerical solution of ~ for various types ·of FEA problems. For the work presented here the Direct Differentiation Method (DDM) is preferred.

Three distinct DDM approaches are discussed, computationally implemented and analysed:

1. 'intuitive' non-shape DSA

2. 'rigorous' non-shape DSA

3. shape, or geometric, DSA

The DDM procedure starts by defining the state of equilibrium at time t + .6..t, t+Ll.tF(u,b), as:

with

t+Ll.tF(u,b) = ~+Ll.tQ(u,b)- t+Ll.t~=O

t+Ll.tQ = 0 - J t+Ll.ts .. bt+Ll.tC' .. odv

0 IJ 0 c-13 ov

(3.2)

for TL,

for UL, (3.3)

so that for the Jth element of a generalised FE discretisation, (3.2) may be formulated as

t+Ll.t FI( u, b) =: ~+Ll.tQ1 ( u, b) - t+Ll.tR1 = 0

Applying the design sensitivity operator 8 = \J b (3.1), the differentiation operation

on t+Ll.t F ( u, b) can be separated into distinct parts:

where, from [35],

- - duj a b =db· 8u·'

I J

The variational expression (3.4) can thus be restated as

8 t+D..t F du · J

auj dbi

(3.4)

37

or

(a t+At F ) . (du ) = -(a t+At F) au I db I ab I

. (3.5)

for the Jfh element of the FE discretisation.

Several assumptions are made which aim to simplify the DSA formulation while still reflecting the majority of actual DSA applications:

• Conservative, or 'non-follower', forces are assumed so that t+~t?R becomes deformation independent and

(a t+Atp) = (a ~+Atq) . au I au I

(3.6)

• In a similar vein, the concentrated nodal traction forces ~+Att, included in t+At?R, have no explicit dependence on the shape design parameters b = h so

that

(at+At3l) =

ab o. t I

• Furthermore, t+At ~ is assumed to have no explicit dependence on the non­shape design variables:

(at+.At3l) -ab - 0 Vb. I

It follows that

(a t+At F) = (a ~+Atq) . ab I ab I

(3.7)

Combining (3.6) and (3. 7), equation (3.5) is reduced to

(a ~+Atq) . (du1) = -(a ~+Atq) . au I db I ~ ab I

The non-shape DDM design sensitivity approaches exhibit complementary charac­teristics:

• The 'intuitive' approach is approximate, but simpler to implement and com­putationally less expensive.

38

• The 'rigorous' approach is completely general and more accurate, but at the same time also more complex and computationally more expensive.

For the purposes of shape DSA, only the 'rigorous' approach is implemented due to its more accurate performance for a wider range of applications.

The section concludes by investigating the applications of DSA to displacement­based reponse variables, specifically stress T.

3.1.2 'Intuitive' Approach

From (2.32) and (2.37) the quasi Newton-Raphson iterative solution scheme can be written generally, for TL and UL, as

fl.Q/k) = t+6t~1 _ ~+6tQ/k-1)'

where fl.Q1(k) is defined in terms of fl.u1 by

fl.Q/k) = K1 · fl.u1. (3.8)

Equation (3.8) suggests that the stiffness matrix K = K 1 may be interpreted as an approximation to the expression in (3.6). Hence, the approximation

8 t+6tQ (k) 8 t+6tQ K= 0 I ~ 0 I

8u1 8u1

may be included in the DDM expression (3.8) so that

(3.9)

The substitution of the stiffness matrix K, obtained from the structural analysis

phase, for 8 ~:~1Q 1 in the 'Intuitive' approach, characterises the central difference

between the 'Rigorous' and 'Intuitive' methods of DSA; this substitution implicitly makes the 'Intuitive' method more approximate than the 'Rigorous' approach. The stiffness K, stored after each structural analysis iteration, can be retrieved for 'the purpose of obtaining the matrix ( ~)

1. The RHS of (3.9) is analytically simplified,

depending on whether shape or non-shape DSA is performed, and discretised.

The 'intuitive' method can be applied to shape and non-shape DSA problems alike. Since the method is by definition approximate, however, it is not applied to shape DSA, or structural optimisation for that matter, in the research presented here: shape DSA is dealt with exclusively in context of the 'rigorous' approach to DSA.

39

3.1.3 'Rigorous' Approach

For the sake of brevity, and particularly with a view towards introducing shape DSA, the TL formulation will be used exclusively in this section.

Employing the definitions in (3.2) and (3.3), and referring to Section 2.1.2, the expression (3.4) is restated for the TL formulation as

8 (j t+t:i.t S·· ht+t:i.te .. odV _ t+t:i.t~) ov 0 ,, 0 ,,

+ b (fov ~+At Sij h~+Ateij oav - t+At~) = 0. (3.10)

In introducing the 'rigorous' approach, it is instructive to generalise the external load expression t+At~ as

t+At~ = J t+t:J.tp 8t+AtU T. t+Atf OdV + J 8t+AtU T . t+Att Odf oy 0 0 0 Oft 0 0 t l

by including body forces in the expression for the externally applied loadings, al­though these forces are neglected for the applications considered in this thesis.

Applying the combined localisation and isoparametric mapping Co ~ C'o ~ Cr, from the global 'initial' frame Co to the 'natural' frame Cr, to (3.10), lJ (t+Atp) in (3.4) is restated as

{f rv b ( 8~+6.te T . ~+l:J.t S J) r dV + f rv b ( 8~+6.te T • ~+t:J.t S J) r dV}

{f rv b (~+t:J.t p h~+t:J.tu T . ~+At f J) r dV + f rv b (~+At p h~+Atu T . ~+At f J) r dV}

{! b ( h~+AtU T. ~+Att Jr) r dft + 1· b ( h~+AtU T. ~+Att Jr) r dft} == Q. rrt rr,

The expression in (3.10) simplifies the DSA procedure subject to several assumptions and conditions, pertaining primarily to the loading conditions. The assumptions in­troduced are standard in much of the literature on DSA; the slight reduction in generality of the DSA formulation is more than offset by the considerable simplific­ation of (3.10):

• Assuming material incompressibility implies

so that effectively

Op_ tp _ t+Atp - 0 - 0

• 40

• The variation of deformation ht+Atu is arbitrary and, therefore, design parame­ter independent:

8 ( b~+Atu) = 8 ( b~+Atu) + b ( b~+Atu) = 0.

• Assuming 'non-follower' forces, ie. the traction t+Att and body force b+At f are not deformation, ~+Atu, dependent:

b (~+Att) = 0, b (~+Atf) = 0 . •

Imposing the conditions above, equation (3.10) reduces to

{Irv 8(8~+AteT.~+Ats) JrdV+ f~v 8(8~+AteT.~+AtsJ) rav}

J b (t+tl.t p 8t+Atu T • t+At f J) r dV -j =g (i5~+Atu T • ~+Att Jr) r dI't ··v . 0 0 0 rrt

= 0 (3.10)

The following specific assumptions are introduced in order to simplify (3.10) for the particular applications considered here:

• Body force is negligible compared to traction, thus

~+At f--+ 0.

• Traction force ~+Att has no shape or non-shape design parameter dependence

b (~+Att) = 0.

• Traction ~+Att is a concentrated nodal force, so that

J 8 (8~+Atu T. ~+Att Jr) T dI't rr,

= [ 8 (15t+Atu T . t+Att)] + [ 8 (15t+Atu T • t+Att)] O O node = 1 O O node = 2

- 0.

Introducing these further assumptions into (3.10), the variational form of the 'rig-orous' DSA formulation may then simply be s}ated as ·

Irv 8 ( 15~+Ate T . t+At s) J T dV +f rv 8 ( ht+Ate T. ~+Ats J) T dV = 0 (3.11)

which applies generally to problems involving shape or non-shape design parameters, subject to the constraints and assumptions imposed above.

The next two sections discuss further simplifications that may be introduced, de­pending on whether shape or non-shape design parameters b are utilised.

41

Shape DSA

In general many real-world structural optimisation applications investigated focus on finding an optimal geometric structural design and, consequently shape DSA plays an important role in computational sensitivity analysis.

Several unified formulations which accommodate both shape and non-shape DSA have been introduced in recent years. The shape DSA procedure employed here is based on a Euler-Lagrange kinematic description first proposed by Haber[16].

Conceptually, shape design sensitivity analysis is implemented by introducing a per­turbation mapping, c; ~ C0 , into Fig.[3.1].

-, ·1

Tl I~

NaturaVParent Frame C r

translation

-x• 1

y• I

LocaVPrinclpal Initial Frame C 0

translation

Figure 3.1: Shape design parameters.

, y I

>

Perturbed Local Initial Frame C~

The perturbation mapping from the original, or 'template', local 'initial' frame c; 2

to the perturbed global 'initial' frame C0 is defined in terms of shape design variables b = bt.

For each truss element the perturbation may be as simple as

X1 = X~ + f:lh1 ,

X2 = x; + f:lh2 ,

for positive or negative structural design perturbations f:lh.

Recalling (2.24) there now exists a clear shape -design variable dependence in the transformation matrix3 otj = 0Tij (fi) i,j = g' Xi,j, and hence, a shape design de­pendence for both the Jacobian integration factors J and Jr. As a consequence of this dependence, (3.11) cannot be further simplified in its variational form for shape DSA.

2Refer to Fig.(3.1] ~T 3

The transformation matrix 0 T now implicitly represents the perturbation mapping Ct -+ Co-+ C'o.

42

Non-shape DSA

A significant portion of the physical problems to which DSA is applied, utilise only non-shape design parameters and thereby avoid the various attendant complications of 'shape' DSA.

Young's modulus and the truss cross-sectional area4 are frequently employed as (non­shape) design parameters for the specific case of the truss/bar element implementa­tion. For the research conducted and results presented here the truss cross-section 0 A was consistently chosen as a representative non-shape design parameter.

Selecting 0 A as a single non-shape design parameter, the DSA variational formulation

in ( 3 .11) simplifies to

f rv b ( 15~+At e T • ~+At S) J r dV + h (o A) Jr L 15~+At e T . ~+.At S J r dL = 0.

3.1.4 Stress Sensitivity Analysis

For displacement based one-field variational formulations, eg. TL or UL, displace­ment is the primary solution variable from which the other reponse variables, for instance strain, are derived. Recalling the form of the sensitivity operators (cf. (3.1) and (3.4)) introduced in Section (3.1.1), the design sensitivity procedures investi­gated thus far can justifiably be referred to as displacement DSA. In the majority of physical applications, however, objective and constraint functions are commonly defined in terms of a combination of reponse variables (displacement, strain, stress). Investigating a generalised stress constraint of the form W = W ( T), the definition of the sensitivity operator in (3.1) is altered as follows:

(3.13)

Even for the relatively simple formulation of a truss structure subject to a stress constraint !P'(r), analytically deriving the constraint sensitivity expression, as for­mulated in (3.12), is a non-trivial exercise, as is demonstrated in the next section.

4Variations in the initial truss cross-section ° A do not influence the initial structural configu­

ration.

43

3.2 Discretisation

3.2.1 'Intuitive' Non-shape DSA

Although the theoretical formulations and concepts introduced for the various DSA procedures in the previous section are relatively straightforward, performing the discretisation of these formulations is somewhat more complex. The discretisations for the 'intuitive' non-shape, 'rigorous' non-shape and shape DSA implementations are entirely original and are generally formulated to enable integration within the user element routines of most FEA codes as 'pseudo'-elements, in particular the UEL module in ABAQUS.

The 'intuitive' approach to DSA is inherently approximate, as mentioned in Sec­tion 3.1.2, and is employed exclusively in applications involving non-shape design parameters for the purposes of the research presented here. In addition, since the 'intuitive' approach is based on the UL formulation, the procedure is usually lim­ited to small strain behaviour, whereas the TL-based 'rigorous' approach is more commonly used for large deformation accompanied by a large strain response.

For the applications considered here, the truss element cross-sectional area A, as a representative non-shape design parameter, was used consistently in generating results for non-shape DSA.

The discretised form of the 'intuitive' approach, for the design parameter A, follows from (2.28), (2.36) and (3.9):

du1 K . db = K . \lb U[ =

which, for the UL, becomes

where

C, = \loAC, = (~)2 { OL t+AtNx,x} { tTxx + E ( t+AtN)}' and for the TL takes on the form

[OTT . o Ao L ( { E o N x,x ® o NT x,x} + { ~Sxx ON}) . OT] . ( 'Vo)+Atu)

- [ 0TT. K · 0 T · ~+Atu] , (3.14)

with

44

3.2.2 'Rigorous' Non-shape DSA

Recalling the (TL) formulation for J (t+.6.t F 1)

J J (8t+.6.t e T t+.6.t s) J r dV + J 6 (8~+.6.t e T • ~+.6.t s J) r dV = 0 rv 0 . 0 rv

(3.14)

from (3.11), and discretising (3.14) in the standard fashion by utilising the isopara­metric mapping so that, recalling (2.28), the following expression is obtained:

J 8~+.6.te T . ~+.6.t s Jr dV = Du T . 0TT . '/(, . 0T. ~+.6.tu. rv (3.15)

Substituting the discretised expression (3.15) into (3.14):

(3.16)

For non-shape DSA, using the truss cross-sectional area 0 A as a representative design parameter, and referring to the RHS of (3.13) for the result of

(3.17)

(3.16) becomes

0TT · [ ( 8K) · 0T · ~+.6.tu + '/(, · 0T · 8 (~+.6.tu)] = - [ 0TT · '/(, · 0T · ~+.6.tu]

which, after a not inconsiderable amount of analytical manipulation, adopts a 'pseudo'­element discretisation

(3.18)

where the 'pseudo' stiffness matrix K' is defined as

K' = 0TT . M . 0T

for

M = ( E o Lo A) [ oN T . oT. ~+iltu. oN + oN ·.~+.6.tu T. OTT. 0) oN + oN. oN T] '

and the 'pseudo' external loading R' may be expressed as

R' = - [OTT . '/(,. oT. ~+.6.tu] .

The ability to formulate the discretised DSA equations in a 'pseudo'-element format is crucially important in enabling implementation of DSA into large FE codes, for example ABAQUS.

45

3.2.3 'Rigorous' Shape DSA

Referring to the analytical formulation of shape DSA in Section 3.1.3, the discretised DSA formulation in (3.16) is rewritten for shape DSA, in a form similar to (3.16), as

(3.19)

where T = T and K, k, are sensitive5 to perturbations in the shape design parameters b = h, ie. T and k, are explicitly dependent on variations of the shape or structural configuration in the mapping C0 --+ C'0 • The difference between (3.16) and (3.19) arising_!'rom the substantially greater complexity involved in implement­ing the operator 8( . .. ) for shape DSA.

The LHS of (3.19) is not affected by the introduction of shape design parameters and, referring to (3.18), (3.19) is therefore simply restated as

(3.20)

The RHS of (3.20), after some manipulation, takes on the form

where, referring to the TL variational formulation, particularly relations (2.27), (2.29) and (2.30), the shape parameter, h, dependent factors in (2.23) are recounted as:

with

0 N x,x + 0N · 0T · ~+L1tu ,

oN T + (!) t+L1tu T. oTT. oN x,x 2 0 .

Evaluating h ( -fc) in (3.21) produces

8(-fc) = 8(0N)·(0A 0i,E) 0~T··+---. 0N·(0A8(0i) E) 0~T

+ •N.(0 A0iE) 6(•f) 5k includes the Jacobian integration factors J and Jr, both of which have an explicit depend­

ency on the it shape design parameters.

46

with

- ( - o~) o N x,x . vb o i, ~ + oN . 6 (or) . ~+tltu

and

(-__;_) o NT . '\] o i, + (!) t+.itu T. s (or T) . oN o L x,x vb 2 o

1 t+tlt T 0 A T 0 A 0 A

- - u · T · N ·'\lb L. oj, o v

Now, applying the explicit sensitivity operator S( ... ) to the shape parameter de­pendent TL transformation matrix6 0 'i' = 0T(h) in (2.25) results in

= (o A ) _ o A _ [O* ] _ [ T 03x3 ] 8 T = \J b T = \J b 0 Xi,j = 03x3 T '

where

[ {<>,cos o+ cos <,6,8,) - {fis} -{a,cos8 + sin,PPo} ] T { ac sin 0 +cos </>l's} {fie} - {as sin 0 + sin </>l's}

· {as} 0 . {ac} for

ac = 8 (cos</>) 'as= 8 (sin</>) ' fie·= S(cos 0) ' l's= 8 (sin 0) .

It remains to derive and calculate the specific form of the expressions

7 ( . O) = d sin 0 o Slll dh

= dcosO 8 (cosO) = dh ,

7 ( . ,1,.) = d sin </> o sm 'I' dh

= dcos </> fJ (cos </>) = dh ' (3.22)

as well as b (0 L ), for each element I of the FE discretisation, in order to obtain the RHS of (3.20). The expressions in (3.22), and h (0 L), are application dependent and can consequently not be formulated generally.

Referring to the pylon truss structure7 in Section 6.4, where the positions of nodes 7-10, and thus the formulation of various elements, are dependent on a single shape design parameter h, these expressions'take on the form summarised in Table[3.1]8•

6 Refer to Section 3.1.3. 7 Refer to Fig.[6.4.3]. 8 Refer to Fig.[2.4]

47

3.2.4 Stress DSA

As stated in Section 3.1.4, many generalised functions, specifically the objective and constraint functionals used in structural optimisation, have a dependence on response variables other than displacement.

The method of derivation of the sensitivity vector for a generalised constraint Vb W

where, for example, W has a stress dependency W = W (r), reduces to the problem of obtaining an expression ~~'where Vb u has already been solved for, and having

analytically derived g;, ~ and ~ in equation (3.12):

(3.23)

The expression in (3.23) reflects the fact that the majority of stress-based constraints 9i prefer to employ the physically measurable Cauchy stress T, instead of the 2nd Piola-Kirchho:ff stress S. Utilising T does, however, significantly complicate the derivation of ~~. . .

Employing the TL formulation, r = t+AtT, is found from the frame transformation operation:

t+AtT (

t+Atp) 0 · T t+At 0 v - -op . t+AtX . 0 S . t+At"'-

0 vT t+AtS 0 X t+At "'- . 0 • t+At

by assuming material incompressibility, and

for

0 x OTT . t+AtT· t+At =

mapping

mappmg C'o-+ Co,

Ct+At -+ C' 0 '

so that t+AtT is dependent on the material deformation ~+Atu.

Referring to (2.31), the 2nd P-K stress for a truss element implementation takes on the form

- T --.. t+Ats = E o.,:-,. • oT . t+Atu 0 xx - ' .i"I 0

and in a· similar vein to the process in (2.35)

t+At7 xx [o x] 2

t+At S - t+At i,j=I o xx

-T <1>2 E oN . oT. ~+Atu,

node: 7 Elements affected: 15, 18, 23

dL _: x-y dh - L

,,

dsin8 _ .=!. _ (-1L-) (£:1l) dh - PL ·PL2 PL

dsin¢ __ (;£) ·(£:.1l.)· dcosO ~ _I _ (J_·) (£:.1l.)

dh - PL PL2 PL

~ _ (£:.1l.) (.1) _ (PL) (=::u) dh- L2 L dh - PL L L2 L

node: 8 Elements affected: 17, 19, 25

\ ..

dsin8 _ .=!. _ (.....JL)· (-(x+y))· dh - PL PL2 PL

dsin¢ __ (;£) (-(x.+y))

dcos8 _ .=!. _ (J_) (-(x+y)) . dh - PL PL2 PL

~ _ (-(x+y).) (.1) _ (PL) (-(x+y)) dh- L2 L dh - PL·· L L2 L

node: 9 Elements affected: 16, 21, 24

dsinB __ I _ (.....JL) (ic£.) dh - PL PL2 PL

dsint/l __ (;£) (ic£.) dh- L2 L

dL _ y-:x . . dh - L

dcosB _· .=!. _ (J_' ·(ic£.). dh - PL PL2} PL .

~ _ (1l.::.=:) ·(.1) _ (PL) (ic£.) dh - PL L L2 L

~}M '0

node: 10 Elements affected: 14, 20, 22

dsin8 __ I _ (.....JL) (~) dh - PL PL2 PL

dsin.p __ (;£) (~) dh - L2 L

dcos8 __ I _ (J_) (~.~ dh - PL PL2 PL

dsin¢ _ (~) (.1) _(PL (~) dh - PL L £2 L

The symbols x, y and z all pertain to node 1. in the parent frame, ie. x = X(i), y = Y(I) and z = Z(1)·

PL= Jx2 + y2

L = J x2 + y2 + z2

Table 3.1: Shape sensitivity factors for the pylon truss structure

48

I 49

where

~ ~ (0 </>, t+At </>, 0(), t+At ())

- cos t+At </> cos 0 </> (cos t+At () cos 0 </> + sin t+At () sin ° ()) ,

so that (~~) · \J bu in the generalised constraint sensitivity equation {3.23) for the

TL truss element formulation, becomes

(

-T ) ao~ OU . \lb u

oT t+Atu • . 0

The form of ;: is easily obtained, having first derived expressions for Bco~:""t<f>, oco~:""16 and osi~:""' 0 , and retrieving the values for similar expressions in terms of 00 and 0 ¢> from storage. Here again, as was the case at the end of Section 3.2.3, the formulation of these expressions is specific to the particular application and can, therefore, not be generalised.

Referring to the 200-bar truss structure investigated in Section 6.3, the expressions listed in the previous paragraph take on the form summarised in Table[3.2].

------ ____________ .;__ ___________________________ ..

50

-I-, -I- are the x-direction nodal displacement U1z U2z

sensitivities for, respectively, node 1 and 2.

8cost+At9 = l ( a2 - 1) OU}z "'I "'12

8sin1+At9 = -;/:

OUlz

8cost+At9 = _l ( a2 - 1) OU2z "'! "'12

8sin 1+At9 _ 9!.f!.. 8u2z - "'13

-I-, r/-- are the y-direction nodal displacement U1y U2y

sensitivities for, respectively, node 1 and 2.

a cos i+.c. 10 ~ _af3 8sint+At9 = l (~ -1)

8u1y - "'13 8u1y "'! "'12

8cos 1+.c.t9 _ 9!.f!.. 8u2y - "'13

8sint+At9 = _l (~ -1) 8u2y "'! "'12

0: = ex1 - tX2)+ (tUI.x - tU2.x) f3 = CY1 - tY2) + Cu1y - tu2y)

1 = Ja2 + (32

where (txi, tyi, tzi) and (tx2, ty2, tz2) are the global coordinates for nodes 1 and 2, respectively.

Table 3.2: Stress sensitivity factors for the 200-bar truss structure

Chapter 4

Structural Optimisation

4.1 Structural Optimisation, Design Sensitivity and Finite Element Analysis

As an extension of the work in Chapter 3 on the formulation and discretisation of various DSA procedures and their implementation into ABAQUS, this chapter goes further by developing a comprehensive optimisation model to be used in conjunction with ABAQUS. This brings to completion two of the primary objectives of this research, namely, the implemention of DSA and structural optimisation into an established FEA code, viz. ABAQUS.

For computational optimisation, particularly in the field of mechanics, the primary focus usually rests on the sensitivity analysis procedure; i.e. the process utilised in obtaining a reliable sensitivity vector \j b 1P is crucial. In the work presented

here this particular sentiment is supported by employing one particular optimisa­tion model, the Pshenichny-Lim-Belegundu-Arora recursive quadratic programming (PLBA RQP) method, throughout. Although computational optimisation theory is an entirely self-contained field of research worthy of investigation, a full analysis and discussion of the PLBA RQP is beyond the scope of this work; the optimisation method purely serves as a vehicle for the practical implementation of various DSA procedures into ABAQUS.

The purpose of structural optimisation is to rrtinitnise a generalised cost functional

f = f(u,bJ = f (u(b),b)

for shape or non-shape design parameters b, subject to a set of generalised equality and inequality constraints

9i = gi(u, b) = gi(u(b), b)

51

52

Commonly f represents the volume of material employed in a structure, or the financial cost as a proportion of the volume of material, whereas the constraints 9i generally reflect displacement, strain or stre~s limits imposed on the structural behaviour.

The approach adopted in the optimisation model employed here is an enhancement of the standard Lagrange multiplier method. A Lagrangian function P (b, µ)is defined in terms of the addition of the cost functional and scalar multiples of the active constraints1

; b and µ,, respectively, represent the design parameters and Lagrange multipliers. The optimisation problem now reduces to a process of minimising the unconstrained Lagrangian function

P(b,µ) = L(b,µ) = f (u,b) + Lµigi(u,b).

' In order to generate a series of iteratively improved approximations to the optimal set of design parameters, b, the optimisation routine requires· values for the first derivatives of the cost and constraint functionals,

for each iterative set of design parameters b(k).

For the computational implementation of structural optimisation, the finite element analysis procedure solves for values of the material displacement u = u(b(k)) for a given set of design parameters b(k). The design sensitivity analysis routine generates the cost and constraint function sensitivity vectors, '?bf and \j bgi, respectively.

The optimisation algorithm implemented here is designed to interface with finite element and design sensitivity analysis algorithms in exactly the fashion described above. As mentioned above, the specific optimisation model employed is based on the extension of Pshenichny's RQP2 linearisation method for constrained optimisa­tion, referred to as the PLBA RQP method, as presented by Lim & Arora (26]; the procedure is described in the following section.

4.1.1 Structural Optimisation with the PLBA RQP

The PLBA optimisation routine is structured intQ...three conceptually separate parts:

• Part I solves for the direction of optimal change p.

1Constraints which are violated for a particular set of design parameters b become 'active'.

Only active constraints are considered in the optimisation procedure. 2 Recursive Quadratic Programming.

53

• Part II solves for the stepsize a in the direction p, thereby establishing the magnitude of the design parameter change.

• Part III employs a variation of the BFGS method to update W, thereby serving to accelerate the optimisation procedure.

Parts I-III are repeated until the variation in the design magnitude a(k)llp(k)ll is less than a set tolerance value.

I - Solving for the vector direction of optimal change p

The family of quadratic programming (QP) optimisation methods proceed by approximating the nonlinear cost and constraint functionals by quadratic func­tions. That is, the nonlinear optimisation problem

minimize J ( u, b)

subject to 9i (u, b) = 0, i E 11

9i ( u, b) ~ 0 , i E 12

where

11 - active equality constraints

12 - active inequality constraints

reduces to the quadratic programming problem (QPP)

minimize

subject to

with

1 T ,.-.,o T 2P . W . P + u . P

G iT 0 . I . p + 9i = 'i E 1

GiT . p + 9i ~ 0 ' i E 12

• p = vector direction of change in design variables that has to be solved for.

• W = symmetric positive-definite approximation to the Hessian of the Lagrangian function

L ( b, µ) = J ( u, b) + L µi9i ( u, b), i E le, le = 11 U 12.

W is iteratively improved with every optimisation step, starting with an initial value of w<0

) = I. Since W is symmetric positive-definite, Cholesky decomposition can be utilised to formulate W in terms of two upper triangular matrices U

54

• G0

and Gi are, respectively, the sensitivity vectors for the cost and con­straint functions, i.e.

I•

G0 - \J bf ( u, b) ,

G' - \J bgi(u, b),

obtained directly from the computational DSA procedures.

Since the objective of the QPP is to solve only for the vector direction, and. not magnitude, of optimal change in the design parameters, the usual QPP normalisation is introduced:

-:;::;;0T 1-T -+G -minimize 2s · s · s

·T .

subject to Cl · s + gi = 0 , i E 11 ·T .

Cl · s + gi :::; 0 , i E 12

where

s 8=-,

Co

- 9i 9i = -·-'

Co Ci

s=U·p

(u-1)T .. Gi (j = ~-.:.---

( l)T · Cj = II u- . G'll,

. by Cholesky decomposition .

The normalised QPP is solved for s in a straightforward manner by first li:r~­earising the problem, using the Kuhn-Tucker conditions, before implementing a variation of the SIMPLEX method to solve the resulting linear optimisation problem.

,----~------···-

• For the purpose of implementing s~ructural optimisation only inequality constraints are considered. Consequently the normalised QPP may be symbolically reformulated, in preparation for the SIMPLEX method, as:·

maximise f (X) = CT · X + XT · D T . X

subject to A· X - B :::; 0

-X:::; 0

,~---- ---- - ----------~---- -·- ------ - ·---------------'II

55

where3

- 1 - -=v - .-1 n [ ·1 T X =.: s, D = -,_I, C = -G , , A = G , B = - [giJ .

• Applying the Kuhn-Tucker conditions the quadratic equations are lin­earised and the SIMPLEX method seeks to construct a suitable basis of non-zero design variables X and multipliers, .\ and h, simultaneously satisfying the equations

with4

-20 T . x + AT • A - h + S* = c A·X+S=B

hT · X = 0 -,xT·S=O

X ~ 0 , .\ ~ 0 , h > 0 , S ~ 0 , S* ~ 0

and a 'pseudo' objective function

S*o = 2::: S*i = 0. I

Having obtained a solution to the normalised QPP in terms of X = s and ,\ = µ, the optimal direction of change p and Lagrange multipliers µ are found to be

II - Calculating stepsize a in the direction p

The vector direction of optimal change of design, p = p(k), is adjusted by a stepsize parameter a = a(k), so that the next iterative approximation b(k+l)

to the set of optimal design parameters is estimated as

b = b(k+i) = b(k) + a(k)p(k)'

with

a(k) - (!)Jk - 2 ' ..

where Jk is the smallest integer q that safufies the inequality . .

3'[]' indicates a matrix, or vector, constituted over all i E 1£ = h.

4 5 and S* are slack variables.

56

Jk is obtained by employing a simple Newton-Raphson line search procedure:

• The estimate 0 < f3 < 1 functions as a 'calibrating' parameter propor­tional to the relative 'flatness' of the functional <P; i.e. /3 :::::::: 0 for \7 b <P :::::::: 0

over some closed domain ~n with n = number of design parameters.

• The 'pseudo' Lagrangian function <P (b) used to determine the stepsize a is defined as

.where

for

f ( u(k)' b(k))

Fe ( U(k), b(k))

R

r

12

t ( u(k), b(k))

F ( u(k), b(k))

cost functional ,

max { 0, 9i ( u(k), b(k)) : i E le = 12} ,

estirp.ate such that R ~ r ~ 0 ,

L lµi I , i E le'

{ i : 9i ( u(k)' b(k)) + £ ( u(k)' b(k))} ' o<k) - F (u(kl,b(k)) ,

- max { 0, 9i ( u<k), b(k))} , i E {all inequality constraints}

and S(k) is a user-defined constraint violation tolerance parameter, e.g. 8 = F defines a violation non-tolerance scenario.

III - Improving the Hessian estimate W

In order to achieve superlinear convergence, the Hessian W, chosen such that W = I for the first iteration k = 0, is revised using the BFGS update formula, as introduced by Lim & Arora [26]. ·

The Hessian approximation W to the Lagrange function L (b, µ,) is obtained from the following procedure, summarised in a stepwise fashion:

• Calculate the vector

whereµ is obtained from the solution of the QPP in part (I).

• The design vector increment is

t = b-b =·ap.

57

• In order to ensure the positive-definiteness of W, the convexity condition t T · q 2: 0 has to be maintained. Hence a convex linear combination of q and W · t is defined

where

with

q' = eq + (1- e) w. t,

1 ' (0.8)T 1- ~ '

~;:::: (0.2)! }

~ < (0.2)T

• The Hessian Wis updated to W = w<k+i)

W=W+a®aT -e®eT

with

q' a= v'f.' e=y';W·t.

This updating procedure is guaranteed to produce a positive-definite matrix W when the original Hessian W is positive-definite, e.g. W = I. The procedure similarly produces a symmetric matrix W when initialised with a symmetric matrix W.

58

4.1.2 The PLBA RQP method algorithm

The PLBA recursive quadratic programming aigorithm is briefly summarised in following stepwise list:

1. Initial state

• Estimate: b(O) E ~n , R(O) > 0 , S(O) > 0 , 0 < (3 < 1 ,

• Set: w<0> = I 2. Search direction

• Determine: F (u(k),b(k)) ::} fe(k) = J2 (k)

• Set: Go(k) = 'V bf ( u(k)' b(k)) ' Gi(k) = 'V b9i ( u<k)' b(k))

• Solve: QPP for p(k), µi (k), i E J/k)

• Check: QP subproblem constraint incompatability ::} expand feasibility region by setting S(k+I) = {!) S(k)

3. Convergence criteria

• Check: F ( u(k), b(k)) , a(k) llP(k)ll < convergence accuracy then EXIT the optimisation routine. ,

4. Penalty parameter

• Calculate: r(k) ' L: lµ~k) I , i E le (k)

• Check: if R(k) ~ r(k) + feasible design then R(k+I) = {!) ( R(k) + r) else if R(k) ~ r(k) +NON feasible design then R(k+I) = R(k)

else if R(k) < r(k) then R(k+I) = 2r

5. Step size

• Calculate: a(k) ::} b(k+I)

6. Hessian update

• Calculate: q<k), e, t<k) * q'(k)

• Calculate: a, e ::} W

7. New iteration

• Set: k = k + 1

• Recurse: Go to step 2

(

Chapter 5

Implementing DSA and Optimisation into ABAQUS

The formulation of design sensitivity analysis (DSA) in order to facilitate the incor­poration of structural optimisation into an established fin1te element analysis code was one of the primary objectives of the research undertaken for the purposes of this thesis. This chapter aims to present a systematic description of the manner in which DSA and structural optimisation were implemented in conjunction with the

I FEA code ABAQUS 5.4.

Including structural optimisation with an established FEA code provides the DSA and optimisation procedures with access to a host of features and facilities in the FEA package~ thereby enabling the application of structural optimisation to a much wider range of problems than would be possible with a small purpose-built analysis and optimisation package.

The various sections of this chapter focus on, respectively,

1. the standard analysis facilities available in ABAQUS and employed in thjs work,

2. the implementation of DSA into ABAQUS, and

3. the structural optimisation procedure us .. ed in conjunction with ABAQUS.

Structural Analysis

ABAQUS is a general finite element analysis program and includes kinematic and material formulations for a wide range of possible applications, including topics as diverse as thermo-mechanics, rock mechanics and electro-magnetism.

59

.. -------~~-r-----r~~~~·""-·•·•-r•"Li~-crri--:in-.i-:i:r~-.:>Up"'.IJ:'7"Urrc-u1-c;J\..,,vl"l1U.T""f'rUp"Cl"UJ'"Cl,l.TCT

displacement files becomes essential. The property files i are used to separately store

(

Chapter 5

Implementing DSA and Optimisation into ABAQUS

· The formulation of design sensitivity analysis (DSA) in order to facilitate the incor­poration of structural optimisation into an established finite element analysis code was one of the primary objectives of the research undertaken for the purposes of this thesis. This chapter aims to present a systematic description of the manner in which DSA and structural optimisation were implemented in conjunction with the FEA code ABAQUS 5.4. .

Including structural optimisation with an established FEA code provides the DSA and optimisation procedures with access to a host of features and facilities in the FEA package, thereby enabling the application of structural optimisation to a much wider range of problems than would be possible with a small purpose-built analysis and optimisation package.

The various sections of this chapter focus on, respectively,

1. the standard analysis facilities available in ABAQUS and employed in this work,

2. the implementation of DSA into ABAQUS, and

3. the structural optimisation procedure us .. ed in conjunction with ABAQUS.

Structural Analysis

ABAQUS is a general finite element analysis program and includes kinematic and material formulations for a wide range of possible applications, including topics as diverse as thermo-mechanics, rock mechanics and electro-magnetism.

59

60

Each simulation is prepared with an input deck which provides a description of the structure to be analysed and the FE facilities to be utilised; the input deck contains information on

• the structure design, location and orientation,

• the loads and boundary conditions imposed,

• the type of element and material model to be used, and

• the step-wise routine outlining the progression of the simulation.

Details of the mesh nodes and element composition are defined in external input files.

ABAQUS makes provision for the implementation of user-defined kinematic models, in conjunction with user-defined material models, via the user-element routine UEL

During each analysis iteration ABAQUS constructs and solves the global element matrices using a N ewton-Raphson procedure employing an optimal wavefront solu­tion algorithm.

The UEL routine

• receives information from the solver for each individual element in the mesh. This information includes the node coordinates, total and incremental dis­placement, the element type, step number, increment number and iteration number;

• constructs the stiffness matrix AMATRX and the internal load vector RHS;

• returns AMATRX and RHS to ABAQUS for assembly into the global stiffness matrix and internal load vector, respectively.

The Total Lagrange (TL) formulation is implemented into ABAQUS in a straight­forward fashion. Implementing the Update Lagrange (UL) formulation, however, requires the U EL routine to access external property and displacement definition files. In order to standardise the TL and UL !)EL implementations, the practice of employing external property and displacement files with the UEL routine was later introduced for the TL formulation; this approach forms the basis for the inclusion of DSA into ABAQUS.

The analysis procedure may be summarised in algorithmic form as illustrated in Fig.(5.1].

Start: Initiate the simulation For: Each step: {

,/ jJ

Define: Boundary conditions and loads

For: Each load increment {

Check: Displacement and Internal Loads convergence

If: No convergence {

For: Each element {

Call: UEL

UEL • Import: Element information from ABAQUS UEL. Set: Construct AMA TRX and RHS UEL. Return: AMA TRX and RHS

Assemble: Global stiffness matrix and force vector

Determine: Solve for displacement increments }

} }

} End: Terminate the simulation

Figure 5.1: The ABAQUS analysis procedure.

Design Sensitivity Analysis

61

In order to implement DSA into ABAQUS, the user element routine UEL had to be suitably altered. A significant component of the research effort contributing to this thesis was focused on developing an appropriate technique for accomplishing this.

The U EL was redefined to enable dual purpose functionality; the user element routine contains code for performing both structural and sensitivity analysis. A flag insi<;J.e the UEL routine was employed to indicate whether the DSA or the structural ana­lysis procedure should be called. This flag was changed whenever a new increment or step was initiated by ABAQUS.; the first increment of every step is reserved for DSA, with the DSA and structural analysis increments alternating strictly for the subsequent increments. Of course, many more-ca.Us are made to the analysis proced­ure for a simple DSA problem: the analysis model is accessed for each iteration of the solver, whereas the DSA procedure doesn't require an iterative solution approach - the discretised DSA formulation is linear in the displacement sensitivities.

62

Constructing UEL so as to convince the ABAQUS solver that it should regard DSA and structural analysis alike depends crucially on being able to formulate the DSA model in terms of 'pseudo'-stiffness and 'pseudo'-load so as to resemble the discret­ised kinematic formulation.

Although the DSA procedure does not utilise an iterative solution procedure, it does require a certain minimum number of iterations to prepare the solver for DSA, to successfully implement sensitivity analysis, and then to prepare for the next analysis increment. For the smallest DSA problem, i.e. single design parameter variation, at least three iterations are required for the DSA increment.

The DSA procedure is subdivided into the following distinct steps:

i. Import and store the response data for the structural analysis increment. Set up AMAT RX and RHS so as to solve for the global load vector.

IL Solve for the global load vector. Set up the stiffness matrix AMATRX a.rtd the internal load vector RHS with the pseudo-stiffness and pseudo internal loads, respectively, in preparation for finding the first sensitivity vector.

m. Solve for the ith sensitivity vector and set up AMATRX and RHS in preparation for finding the ( i + 1 )th sensitivity vector. This process is repeated n - 1 times, where n represents the number of design parameters for which response sensitivities are obtained.

iv. Solve for the nth sensitivity vector and set up AMATRX and RHS to produce a zero displacement increment, thereby halting the DSA increment.

A 200 design parameters sensitivity analysis problem therefore requires 202 itera­tions of the DSA increment. From the DSA increment outline enumerated above, it is apparent that much work during the sensitivity analysis phase is involved with the construction of suitable RHS 'load vectors' for each iteration.

Preventing ABAQUS from unnecessarily subdividing the incremental time steps upon encountering hundreds of DSA iterations requires a careful setting of the ABAQUS solver parameters; ABAQUS provides for this eventuality with the CON­TROLS keyword.

A similarly precise calibration of the solver is necessary when analysing a problem likely to yield very small sensitivities: small sensitivity values often cause the DSA increment to terminate prematurely by meeting the displacement convergence cri­teria, e.g. the DSA increment may halt after obtaining 166, of 200, sensitivity vectors if the response sensitivity to the l66th design parameter is smaller than the solver's displacement tolerance.

For the purposes of performing DSA in ABAQUS, the use of external property and displacement files becomes essential. The property files· are used to separately store

63

information important to the structural analysis and DSA procedures. The U EL routine keeps track of the correct total and incremental displacements, as well as the virtual displacements contributed by the various sensitivities and the global load vector.

Employing the same variable array to solve for displacement, displacement sensiti­vity and global loads results in a loss of significance for small sensitivity values. This effect is noticeable in the sensitivity results for small load cases by manifesting as a divergence in FDM and DSA estimates. The nature of the DSA implementation in ABAQUS furthermore imposes a theoretical limit on the magnitude of the largest load cases which can be dealt with by the analysis procedure - structural load cases in excess of 1000 kN have been accommodated successfully.

The DSA procedure incorporated into UEL can be represented algorithmically as illustrated in Fig.[5.2].

Structural Optimisation

The PLBA structural optimisation routine is not integrated into ABAQUS, but works in conjunction with ABAQUS as a separate executable module:

• ABAQUS completes a full analysis - alternating between structural and sen­sitivity analysis - based on a given structural design and fixed set of material parameters.

• On completion of the ABAQUS analysis the optimisation module is called. The optimisation routine utilises the structural response and sensitivity values generated by the preceding analysis to estimate a more optimal set of design parameters.

• The structure configuration and material parameters in the property files and other data files are updated accordingly. Subject to the optimality convergence criterion not being met, a new analysis is initiated.

The optimisation cycle is implemented as a operating system script file: in order to allow ABAQUS to smoothly integrate with the optimisation sequence, the standard ABAQUS script files was rewritten.

In addition to the property files utilised by the analysis and DSA module, data files were introduced to store the response and sensitivity information generated by the analysis/DSA module as well as the design parameter variations generated by the optimisation module.

64

The four modules or elements of the optimisation cycle may be enumerated as follows:

1. The ABAQUS 'PRE' executable processes the information in the input deck and various input files and assimilates the data, written to a data file, into a form accessible by the ABAQUS analysis/DSA module.

2. The ABAQUS analysis/DSA module aq::esses information in the 'PRE' data file and additional, external property files. The response and sensitivity vectors are calculated and the property and sensitivity information files are updated accordingly.

3. The optimisation module inputs information from the sensitivity information file, performs an optimisation and writes the improved design parameters to the optimisation information file. For the purposes of the research presented here, the objective and constraint functions, and their gradient formulae, were included in the optimisation module. Ideally the optimisation procedure would be made more general by including the application specific details in a separate source file.

4. The design update module reads the optimisation information file and tests for convergence of the design parameters to withfn tolerance. ff the convergence check fails, the property and node definition input files are updated. In a sim­ilar fashion to the optimisation module, the design update module is specific to the problem being investigated.

The optimisation cycle is graphically illustrated in Fig.[5.3] by means of a flowchart.

Start: UEL is called

}

Import: (From property files) Property, solution variables (From property files) Element counter Element information from ABAQUS

If: First iteration, first increment, first step, first element ? {

} Set: Procedure flag to DSA

If: New Increment or new step; old iter. value > 1 {

If: First element and procedure flag is set to ANALYSIS {

} Set: Procedure flag to DSA

If: First element and procedure flag is set to DSA {

} Set: Procedure flag to ANALYSIS

}

If: Procedure flag is set to ANALYSIS {

}

Import: (From property files) Total and virtual displacement Set: Construct AMATRX and RHS

If: Procedure flag is set to DSA {

}

If: First iteration {

}

Export: (To property files) Response variables Set: Construct AMA TRX and RHS to find global load

If: Iteration NOT first and NOT last {

Import: (From property files) Old and virtual displacement Export: (To property files) New displacement ·

If: First sensitivity iteration {

} Else: {

}

Determine: Global load Set: Construct AMA TRX and RHS to find first

sensitivity vector Export: (To property files) Global load and virtual

displacement

Import: (From property files) Global load Determine: Ith sensitivity vector Set: Construct AMA TRX and RHS to find (i+1)th

sensitivity vector Export: (To property files) Virtual displacement

If: Last sensitivity iteration {

}

Import: (From property files) Old and virtual displacement Determine: Last sensitivity vector Set: Construct AMA TRX and RHS to produce zero

displacement increment. Export: (To property files) Virtual displacement

Export: (To property files) New property, solution variables

Return: AMATRX and RHS End: ExitUEL

Figure 5.2: Implementing DSA into ABAQUS.

65

Setup

Properties

ABACUS

PRE.1

Opthnlsatlon

Update

ABACUS DlllFlll

l'ro!llrtll• Flies

Figure 5.3: The optimisation cycle.

66

Chapter 6

Numerical Results

6.1 The 3-bar truss

Introduction

The 3-bar truss, due to its geometric simplicity, is generally used as a benchmark for testing different design sensitivity formulations; the structure investigated here is similar to examples in [31] a.nd [32]. The results presented in this section are intended to demonstrate

i. The implementation of single-parameter DSA in ABAQUS.

ii. The performance of the 'intuitive' approach to non-shape displacement DSA.

Figure 6.1.1: 3-bar truss configuration.

Material and structural design parameters, as well as loading and boundary condi­tions, are listed in Table[6.l.l].

67

68

I

Results Summary

The following results were produced for the 3-bar truss structure:

1. Graphical results for a representative loading and deformation of the 3-bar truss is presented in Fig.(6.1.2].

2. Tables(6.l.2-6.l.3]1 present comparative results,· verifying the design sensitivity and element formulations, for physically realistic values of the design parame­ters band applied load P, i.e. the values were chosen so as to produce element strain of less than 5%.

3. A selective comparison of the FDM and DSA procedures in estimating the displacement sensitivities ~ over a range of design parameter values b, and

J

applied loads P, is presented in Figs.(6:1.3-6.1.6].

4. Figs. [6.1. 7-6.1.8] similarly provide comparisons between the kinematically non­linear UL, ABAQUS linear and ABAQUS nonlinear element formulations, for a range of design parameters and applied loads.

2

Figure 6.1.2: Initial apd Deformed truss structure.

• The CPU time for UL includes both structural and sensitivity analysis phases. All simula-tions were performed on an IBJVI RS/6000 370 workstation. .

NL = • LIN =

UL = Nonlinear kinematic formulation Linear kinematic formulation Updated Lagrange NL formulation

68

- Young's modulus - Element cross-section - Structure proportions

- Load

- Constraint

- UL kinematically nonlinear

Physical properties

E = 10000 ksi b; = A = 0.01 in2• i = 1. .. 3 · cf. Fig.[6.1.1)

Loads and Constraints

69

P acts at an angle () = 45° below the horizontal at node point 1, as illustrated in Fig.[6.1.1) Node points 2, 3 and 4 are fixed.

Analysis procedure

DSA procedure

- Sensitivities calculated with regard to non-shape cross-sectional design parameters b utilising a displacement-based 'intuitive' DSA approach.

(j)

-Calculat[e(d~~J i)n 2

orderl to obtain the objective functional sensitivity Vb W~il, where

w~il = o.~05 - 1 , so that

with j = node number and i = x or y

Table 6.1.1: Analysis data.

70

Results

For a particular selection of design parameter values, b, and applied load P, Table(6.1.2] is intended to provide a comparison of the functional sensitivity vectors ~: genera­ted by the DSA and FDM procedures. Table(6.1.3] similarly serves to investigate the relative performance of the UL analysis procedure employed in this section. The results presented in Tables[6.l.2-6.l.3] were obtained for physically realistic strain response values of 1.3%-3.9% - i.e. the strain is sufficiently small for the UL proce­dure to be viable - and consequently a better than 97% DSA-FDM correspondence is obtained in most instances.

dllf(l) x

dllf(l) x

dllf(l) x

dllf(l) _ Y_

dllf(l) _Y_

dllf(l) _Y _

db1 db2 db3 db1 db2 db3

DSA -227880.7 313579.5 -292793.6 -53200.9 -217816.8 130878.9 FDM -233476.1 299964.7 -298593.7 -56190.6 -215479.2 133479.9

DSA/FDM 97.6% 104.5% 98.1% 94.7% 101.1% 98.1%

b1 = 0.1 in2 , b2 = 0.01 in2

, b3 = 0.01 in 2 , P = 40 kips , E = 104 ksi

Table 6.1.2: DSA vs. FDM - Comparing sensitivities at node 1

Disp.: Node 1 UL

0.5389 -0.2656

2.23

ABAQUS NL ABAQUS LIN

0.5392 -0.2666

0.25

0.5432 -0.2250

0.17

UL/NL

99.9% 99.6%.

b1 = 0.1 in2 , b2 = 0.01 in 2 , b3 = 0.01 in 2 , P = 40 kips , E = 104 ksi

Table 6.1.3: Verifying the UL element formulation

_J

I 71

The performance of the 'intuitive' DSA approach is compared to that of the FDM in Figs.[6.1.3-6.1.6] for a wide range of design parameter, b, and load, P, values. In most of the figures a distinctive divergence. of the FDM and DSA sensitivity estimates are noted for large deformation accompanied by large strain.

' . . . . -100 '

' -400'

,,... llr•ln In elem. 1 , • 12.6'4 lenalle

Sensitivity: !.l}'. to

E • 10000 kal P • 40 klp1

b2 • 0.01 b3 • 0.01

- OSA for ILi

-+ FOM for Ill

~ OSA for Ult

- K- · FOM for Ult

120

80

40

-t:_~~~~~s:t:====--e---1!1-B-ft---fl1--~-e-~~o ~ ~ 1 ro

Cross-sectional design parameter bl

. . . du1 . d . t b Figure 6.1.3: Displacement sens1t1v1ty dbi vs. es1gn parame er 1·

Ors~en~s~if~1v~it~Y=~!lx==t~o~b.2#=~--1f--~~+-+-se_n_s_iti_vi~ty~:-!.l}'...._to~~200

-150

I

' :-200.:

E • 10000 bl P • 40 klp1

b1 • 0.01 b3 • 0.01

•train In elem. 2 • 8.8'4 lenalle

strain In elem. 2 • 16.5'4 tensile

- OSA for ILi ·+· FDM for Ill ~ DSA for Ult

· *· FDM for \!.l!

150

100

50

-250L-~~~~....::::9:!::====111---i:1-1a-e1---l31--~-a-~~o M1 ~ 1 ro

Cross-sectional design Pilrameter b.2

D. I . . . du1 d . t b Figure 6.1.4: 1sp acement sens1t1v1ty db2

vs. es1gn parame er 2·

-10

-20

-30

E • 10000 kal P • 40 klpa

b1 • 0;01 b2 • 0.01

-:- OSA for Ux -+ FOM for llJl

·40 :' J atraln In elem. 3 ··6.2" compreaalv• (max. atraln : 31.6" tenalla In elem. 1)

-a- OSA for Ux

•*· FOM for Ill!

0.01 0.1 1 Cross-sectional design parameter b,3

,<

-10

-20

-30

-40

10

Figure 6.1.5: Displacement sensitivity ~~: vs: .d.~sign parameter b3 .

Sensitivity: !J.! to !!] Sensitivity: !Jl! to bJ Ot=~='1==J:==::=+:=:+:::::::;::--:----~-:--,----i1e

14 -10

max. atraln : 14.1" tanalle In elem. 1 12

- 20 -.., DSA for llJl 10

-+ FDM for I.ix -a- DSA for U:t 8

-30 - l<- FDM for Ill! 8

bl • 0.1 , I • 1. .. 3 E • 10000 kll

-40 max. strain: 7.1 .. tenalla In elem. 1 4

2

-501b=::B=:ii~==:!:::::!:==----------,_J 0 4 40 400

Applied loaq e

F. o· I . . . . dui 1· d I d P 1gure 6.1.6: 1sp acement sens1t1v1ty -.d vs. app 1e oa . b1 . .

72

I 73

In verifying the relative performance of the UL analysis procedure, cf. Figs.[6.1.7-6.1.8), for the same range of design parameter b2 and load values employed in the sensitivity comparisons, a similar trend of decreasing accuracy with increasing strain is noted.

As illustrated in Fig.[6.1.7), the ABAQUS linear bar element formulation fails inex­plicably for the entire range of strain values employed in this particular applications. Several modifications were made to the problem in an attempt to obtain a sensible response from the linear element, without success.

-- Updated Laorange: lbl

· + · ABACUS Nonlinear: !bl

0 E • 10000 kll P • 40 klpa

· 0 · ABACUS Linear : !bl

--*- Updated Lagrange: U.X

· <} · ABACUS Nonlinear: U.X

·I:!.· ABACUS Linear : U.X b1 • 0.01 b3 • 0.01

0.01 0.1 1 Cross-sectional design parameter b2

-2

-3

10

Figure 6.1.7: Displacement u 1 vs. design parameter b2.

5 max. atraln : 7.1llo In elem. 1

4 - Updated Lagrange: lbl

+- ABACUS Nonlinear: !bl

3

2

~- ABACUS Linear : !bl

""*- Updated Lagrange: U.X

-<) - ABACUS N onllnear: U.X

-L!.- ABACUS Linear : U.X

bl • 0.1 , I • 1. .. 3

max. atrain: 14.1llo in elem. 1

o.k=lB=:l!i:=:!l!=:~:::!:::::'.~-=---------_J -2.s 4 40 400

Applied loading e

Figure 6.1.8: Displacement u 1 vs. applied load P.

Results discussion

.. ,!J

/)

74

• The marked increase in computational time for the user UL element formu­lation, noted in Table[6. l.3], is largely due to the inclusion of the sensitivity analysis procedure in the ABAQUS user element routine UEL. For multi­parameter DSA, however, the computational time increases at a .less than linear rate with respect to the number of design parameters, as is evident from Table[6.2.2] in the next section. Computational costs for truss-like structures remain well within reason.

• The notable divergence of the FDM and DSA estimates. for large strain, or rather, the lack of such divergence for large deformation· accompanied by small strain in Figs.[6.1.4-6.1.5], is directly attributable to the decreasing accuracy of the UL formulation for large strain. This results from the fact that the 'intuitive' DSA approach depends on the stiffness matrix produced py the UL analysis procedure and the large strain performance of the UL, hence, detrimentally influences the sensitivity analysis routine. This issue is further underscored by results presented in the ensuing sections.

• Of interest is the y-direction displacement behaviour of nodal point 1 for vari­ation in the cross-sectional design parameter b2, cf. Figs.[6.1.8]: for increasing values of b2, element 2 starts behaving like a rigid body and the truss tends to perform more like a pendulum and less like a deformable body, hence the transition from negative to positive y-displacement.

\

,, /J

75

6.2 The 10-bar truss

Introduction

Extending the application of the ABAQUS-integrated DSA implementation to the 10-bar truss problem, cf. Fig.[6.2.1), the aims of this section are

i. primarily to demonstrate the feasibility of multi-parameter DSA with ABAQUS.

ii. to verify the performance of the displacement-based 'intuitive' DSA approach for the more complex 10-bar truss structure.

The truss structure investigated here is broadly based on similar examples in [5] and [32]. .

II II 36 36

2

6 36 II

' Figure 6.2.1: 10-bar trus .. s configuration.

Details pertaining to the material properties, loading conditions and the DSA pro­cedure are summarised in Table[6.2.l).

Results Summary

The following results were produced for the 10-har truss structure:

1. Fig.[6.2.2] graphically illustrates a typical structural deformation for a repres­entative loading condition.

2. Table[6.2.2] provides a comparison of displacement estimates u:i:, Uy at node 1 for the loading illustrated in Fig.[6.2.2].

3. In Table[6.2.3] a comparison of FDM and DSA displacement sensitivities ~~} is presented.

4. Figs.[6.2.3-6.2.4] contain a comparative evaluation of the DSA and FDM dis­placement sensitivities ~ for a range of values of bi, and P, respectively.

5. Figs.[6.2 .. 5-6.2.8] conclude with a verification of the UL analysis procedure: the displacement u = u1 at node I is monitored for a range of values of the cross-sectional design parameter b1 , and load P.

Results

The UL kinematically nonlinear formulation is used here, as with the 3 and 200-bar truss structures, largely due to its compatibility with the relatively straightforward 'intuitive' DSA approach. As a consequence the loading condition is limited to phys­ically realistic scenarios, i.e. strain< 5%. Fig.[6.2.2] illustrates a typical deformation mode.

5

6

Figure 6.2.2: 10-bar truss structure loading and deformation.

76

- Young's modulus - Element cross-section - Structure proportions

- Load

- Constraint

- UL kinematically nonlinear

•

Physical properties

E = 10000 ksi

/J' //

. bi = A = 0.1 in 2 , i = I ... IO

cf. Fig.[6.2.1]

Loads and Constraints

77

. P acts at an angle () = 45° below the horizontal at node points 4 and 2, as illustrated in Fig.[6.2.lJ Node points 5 and 6 are 'fixed.

Analysis procedure

DSA procedure

- Sensitivities are calculated with regard to non-shape cross-sectional design parameters b utilising a displacement-based 'intuitive' DSA approach.

(j)

-Calculat[e(d~~) ~n ordler to obtain the objective functional sensitivity V'b w~i), where

w~i) = u; - I , so that

with j = node number and i = x or y

Table 6.2.1: Analysis data.

,! //

78

Table[6.2.2J supports Fig.[6.2.2] in, numerically comparing displacement values at node 1 for the Total (TL) and Updated Lagrange (UL) procedures, employed here for analysing the IO-bar truss, with the ABAQUS linear and nonlinear (102 ele­ment formulation. The table clearly demonstrates the decreasing accuracy of the UL procedure for increasing values of strain, hence, for large strain problems, the TL is adopted along with the more complex 'rigorous' DSA approach. The table addition­ally reveals that the computational time increases two to three-fold for the UL and TL user element routines; this increase is directly attributable to the incorporation of DSA into the ABAQUS user element routine UEL.

Disp.: Node 1 TL UL ABAQUS NL ABAQUS LIN TL/NL UL/NL

Ux 0.959 1.140 0.956 1.045 100.3% 119.2% Uy -3.681 -3.745 -3.442 -3.606 106.9% 108.8%

CPU secs. 2.640 1.920 0.870 0.800 303.4% 220.6%

bi = 0.5 in 2 , i=l ... 10 P = 100 kips

Table 6.2.2: Node 1 displacements Ux, uy for a representative loading condition . . The incorporation of multi-parameter DSA in ABAQUS is a crucial step towards enabling multiple constraint optimisation. Table(6.2.3J compares the FDM and DSA

y;(ll objective function sensitivity vectors 8 ab evaluated at node 1. From the table the satisfactory performance of the DSA procedure for large sensitivities is evident, however, for very small sensitivities a notable divergence is observed (cf. b5).

Figs.[6.2.3-6.2.4] serve to verify the displacement-based 'intuitive' DSA approach. The DSA and FDM displacement sensitivities ~' and ~~, are compared for a wide range of variation in the values of the cross-sectional design parameter b1 = A, and the applied load P, respectively. For this particular truss configuration and loading condition, DSA-FDM convergence is achieved for deformations well exceed­ing physically realistic values, underscoring the robustness of the DSA formulation.

l' I IJ

d\J! (I) .6. lJ!(l) DSA

dlJ!(l) .6. lJ!(l) Des. var bi _x_ x ._Y_ y

dbi .6.bi Fl5M dbj .6.bj '

bl -l.361E-1 -l.404E-1 96.9% -7.629E-1 -7.872E-1 b2 -2.553E-2 -2.523E-2 101.0% -5.331E-2 -5.151E-2 b3 -2.173E-3 -2.916E-3 74.5% -1.152E-1 -1.216E-1 b4 2.362E-3 2.916E-3 80.9% 4.055E-2 4.383E-2 b5 -7.068E-5 -3.246E-5 217.7% 2.518E-5 3.653E-4 b6 2.664E-3 3.132E-3 85.1% 4.324E-2 4.566E-2 b7 -3.399E-2 -3.769E-2 90.2% -5.417E-1 -5.783E-1 b8 2.013E-2 1.815E-2 110.9% -2.870E-1 -2.988E-1 b9 -l.629E-2 -1.847E-2 88.2% -2.641E-1 -2.779E-1

blO 8.038E-3 7.561E-3 106.3% -1.421E-1 -1.410E-1

bi = 0.5 in 2 , i = 1 ... 10 P = 40 kips

Table 6.2.3: DSA vs. FDM - Comparing sensitivities ~~.1 • . '

;( ·~.

I'

DSA Fl5M

96.9% 103.5% 94.7% 92.5% 68.9% 94.7% 93.7% 96.0% 95.0%

100.8%

79

The large strain behaviour, observed in Fig.[6.2.3] for the variation in bi, is partic­urly illuminating - the FDM/DSA sensitivity response can be separated into three categories:

8u1 8u1 i. The large strain region, for very small values of bi, where - ~ -8b1 DSA Ob1 FDM.

ii. The small sensitivity domain where

8u1 and 8u1 < 10_4

obi FDM obi DSA

OU1 OU1 -I

but often !:lb ob ~ 1 (or~ 1). u I DSA I FDM

iii. The sensitivity domain, excluding those regions listed in the preceding cat-egories, for which ~

1 > 8u1 and 8u1 > 10-2 - 8b1 DSA Obi FDM -

'

8u1' '8u 1

-i and usually a 8b :::::: 1 b1 DSA 1 FDA!

-100

-150

f 1 + FDM-DSA convergence: max. atrafn "4.9 .. In elem. 7 i bl• 0.1 IQf. In .• I• 2-.10

+- - DSA for !.l.K

·+ FDM for !.l.K

-& DSA for llJf

- K- FOM for llJf

E • 10000 hi P • 40 klp1

FOM-DSA convergence: max. &train "4.9 .. ln·elem. 7

300

200

100

-200'-~~~~~~~~..i=== ......... -e--a-~-a-~--i!l----e:O 0.01 0.1 10

Cross-sectional design parameter !21

Figure 6.2.3: DSA and FDM estimates ~~: vs. b1

-2 FOM-DSA convergence: max. strain 1.3'4 In elem. 7J

bl• O.S 1qr. In .• I• 1 ••• 10

DSA for !.l.K -4 --f- FOM for I.I.I

-& OSA for llJf ·X- FOM for ll:i

-6 E • 10000 kll

FOM-DSA convergence: max. strain 1.3ll. In elem. 7

1 -8

15

10

i!i-~...,._--~=a:==~=====~~~~~~..J.o 0.4 4 40 400

Applied load e

. du1 Figure 6.2.4: DSA and FDM e.st1mates dbi vs. P

80

81

The displacement response u = u1, at node point 1, is compared in Figs.[6.2.5-6.2.8] for various element formulations over the same range of values b

1 and P used

in Figs.[6.2.3-6.2.4]. The comparison serves to: ,,

1. Underscore the relationship between sensitivity response and strain.

11. Confirm the UL analysis results and strain behaviour for large structural de­formation of the 10-bar truss.

E • 10000 kll P • 40 klpa

6 - •- Updated Lagrange: I.lg

-+- Total laprange: I.lg

-B- ABAQUS Nonlinear: I.lg

~ ABAQUS Linear: ILi 4

bl • 0.1 aqr. In •• I • 2 •• 10

2

ol_~~~~-==~~~~~~ 0.01 o~ 1

Cross-sectional design parameter bl 10

Figure 6.2.5: Node 1 displacement u~ vs. b1

Displacement: U:t: at node J

-5

E • 10000 kll I' • 40 klpa

-10 - Updated Lagrange: U.X -+- Total Lagrange: U.X - 0- ABAQUS Nonlinear: Uy

-"*- ABAQUS Linear: Uy

-15 bl • 0.1 lqr. In. , I • 2 •• 10

-20'--~~~~~~~~~~~~~~~~~~~~---l 0.01 ~1 1

Cross-sectional design parameter b.l 10

Figure 6.2.6: Node 1 displacement u31

vs. b1

6

5

4

3

2

· +- Updated Lagrange: Ill

Total Lagrange : Ill

-a- A BAO US N onllnear: Ill

--*" ABAOUS Linear : \IA

E • 10000 hi bl• 0.8 aqr. In., I• 1 ... 10

P • 100 kips: max. strain 2.5% In elem. 7

,+ ,'

OIBll';..._~~~~~~~~~~~~~~~~~~~~~~-'

-5

-10

4 103 202 Applied loading e

301

Figure 6.2.7: Node 1 displacement Ux vs. P

Displacement: ~ at node 1

,- P • 100 klp1: mu. •1raln 2.5% In elem. 7

.,_ _ ,r P • 150 klp1: max. strain 3.7% In elem. 7

--

E • 10000 kal bl• 0.8 aqr. In •• I• 1 ... 10

---9

-+- Updated Lagrange: U.X

-15 Total Lagrange : Ill!

- El- ABAQUS Nonlinear: U.X

--*" ABAQUS Linear : U.X

-20'--~~~~~~~~~~~~~~~~~~~~~~--'

400

4 103 202 Applied loading e

301 400

Figure 6.2.8: Node 1 displacement Uy vs. P

82

83

Results discussion .

• For very small sensitivity response values.(< 10-4 ) numerical errors, largely attributable to a loss of significance in the ABAQUS solver (which in the particular DSA implementation employed here is used to resolve sensitivities of order 10-4 amid numerical values often exceeding order 104), cause inac­curacies in the DSA sensitivity estimates, cf. b5 in Table[6.2.3). Due to the relatively negligible magnitudes of these sensitivities, such inaccuracies have no observable negative influence and are generally disregarded.

• The divergence between FDM and DSA sensitivity response estimates, partic­ularly noticeable in Fig.(6.2.3), is a result of the tendency of the UL procedure to overestimate structural deformation for large strain (cf. Figs.[6.2.5-6.2.8)) and, hence, adversely affects the stiffness matrix on which the 'intuitive' DSA approach is based; this causes the DSA procedure to become progressively more inaccurate. The TL formulation, used in conjunction with the 'rigorous' DSA approach, in most cases proves to be much more effectual for large-strain deformation cases.

• The load magnitude and orientation, as well as material parameters, for which displacement results are presented in Figs.[6.2.5-6.2.8), were chosen so as to maximise the cumulative strain effect in the structure. This effect is clearly visible in the divergence of the UL procedure with respect to the other element formulations. The aim in doing this is to demonstrate the non-dependence (on the element formulation) of the functional accuracy of the 'intuitive' DSA approach; i.e. even in instances where the ·reponse variables are inaccurate, the convergence of FDl'v1-DSA results, based on thes~ variables, is nevertheless high in most cases, cf. Figs.[6.2.3-6.2.4).

• Failure of the UL element formulation for very large strain values is apparent in Figs.[6.2.5-6.2.6] for b1 ~ 0.01.

84

6.3 The 200-bar truss

Introduction

The results presented in this section were generated for a 200-bar truss structure Fig.[6.3.l] based on the example investigated in [5] and.[32].

Figure 6.3.1: 200-bar truss configuration.

The elements in the truss structure are apportioned into 29 groups, each of which is assigned specific physical properties and loading conditions [5]; refer to Table(6.3.l].

The physical properties of the truss structure, the loading and boundary conditions imposed, as well as the sensitivity analysis procedure employed, are summarised in Table(6.3.2]. ·

The objective of this section is two-fold:

1. To demonstrate the feasibility of extending the DSA procedure, utilised in the previous results section, to accommodate generalised objective and constraint functionals, i.e. functionals W = W (u, ~, u), dependent on a range of response variables.

2. To underline the capacity of the DSA procedure to perform multiple parame­ter sensitivity analysis, as first demonstrated in the 10-bar results section. Consistently selecting the bar element cross-section as the non-shape design parameter, the sensitivity of displacement u at node 1 is calculated subject to variation in all 200 bar element's cross-sections, thus producing a 200 x 2 sensitivity vector ~~. This result is crucial in illustrating the value of DSA, and its stable implementation into ABAQUS, in enabling the implementation of complex, multi-parameter optimisation.

Additionally, in keeping with the approach adopted in [32], the DSA procedure is complemented by a routine which identifies the maximum stress element for each of 29 element groups (cf. Table[6.3.l]) and performs stress sensitivity calculations exclusively for these 29 elements; a 200 x 29 stress sensitivity vector is obtained.

Results Summary

The following results are presented in this section:

1. Graphical results for a representative loading and deformation of the 200-bar truss are presented in Fig.[6.3.2].

2. Table[6.3.3] presents a comparative results sampling, verifying the sensitivity formulation for physically realistic values~ of--b and applied loads Px, Py.

3. Table[6.3.4] contains the results .of a 200 parameter multivariate DSA per­formed for the displacement u at node point 1.

4. In Figs.[6.3.3-6.3.4] a linear regression of FDM vs. DSA results, for the sensit­ivities ~~}, is performed as a verification of the DSA displacement sensitivities presented in Table[6.3.1]; the design para.meters employed in the comparison

85

Group number Elements

1 1-4 2 5, 8, 11, 14, 17 3 19-24 4 18, 25, 56, 63, 94, 101, 132, 139,

170, 177 5 26, 29, 32, 35, 38 6 6, 7, 9, 10, 12, 13, 15, 16,

27, 28, 30, 31, 33, 34, 36, 37 7 39-42 8 43, 46, 49, 52, 55 9 57-62 10 64, 67, 70, 73, 76 11 44, 45, 47, 48, 50, 51, 53, 54,

65, 66, 68, 69, 7i, 72, 74, 75 12 77-80 13 81, 84, 87, 90, 93 14 95-100 15 102, 105, 108, 111, 114 16 82, 83, 85, 86, 88, 89, 91, 92,

103, 104, 106, 107, 109, 110, 112, 113 17 115-118 18 119, 122, 125, 128, 131 19 133-138 20 140, 143, 146, 149, 152 21 120, 121, 123, 124, 126, 127, 129, 130,

141, 142, 144, 145, 147, 148, 150, 151 22 153-156 23 157, 160, 163, 166, 169 24 171-176 25 178, 181, 184, 187, 190 26 158, 159, 161, 162, 164, 165, 167, 168,

179, 180, 182,\183, 185, 186, 188, 189 27 191-194 ~

28 195, 197, 198, 200 29 196, 199

Table 6.3.1: Grouping information for the two-hundred bar truss.

86

- Young's modulus - Element cross-section

groups 1-12, 14, 16, 17, 19, 21, 22, 24, 26

groups 13, 15, 18, 20, 23, 25

groups 27-29 - Structure proportions

Physical properties

E = 10000 ksi

A= 2.24 in 2

A= 7.07 in 2

A= 14.14 in 2

cf. Fig.[6.3.1]

Loads and Constraints (P =Pc =Py)

- Load 1

- Load 2

- Constraint

- UL kinematically nonlinear

.. Pc acting in the positive x-dirn. at nodes 1, 6, 15, 20, 29, 34, 43, 48, 57, 62, 71 Py acting in the negative x-dirn. at nodes 1-6, 8, 10, 12, 14-20, 22, 24, .... 71-75 Nodes 76, 77, are fixed.

Analysis procedure

DSA procedure

- Sensitivities calculated with regard to non-shape cross-sectional design parameters b utilising a stress-based 'intuitive' DSA approach.

C I ·1 oau . d b . h b" . f . I . . . aw i h - a cu ate ab rn or er to o tarn t e o Ject1ve unct1ona sens1t1v1ty ab , w ere

IP;= [ (;~~)' -1]. so that

\J!. _ diI! i _ diI! i ( [aa11] T . du 8a11) ai!J 'Vb J - db - Alan au db + ab + ~

= ~(8a1~)T. (du) 20000 au db

with j = element for which a 11 is measured.

Table 6.3.2: Analysis data. 87

88

are bi for i = 11, 32, 49, 70, 87, 108, 125, 146, 163, 170-177, 184, 200. The loads Px, Py are a uniform 200 kips and the element cross-sections are as defined in Table(6.3.2].

5. Figs.(6.3.5-6.3.6] compares the FDM and DSA generated sensitivities of the stress-based objective functional '11 199 , chosen as dd<llb 199 for simplicity, subject

200 to variation in the design parameter b200 and the load magnitudes Px, Py, respectively.

6. In conclusion Figs.[6.3.7-6.3.8] present the comparative values of stress a11 in element 199 produced by the UL procedure implemented here, as well as the ABAQUS kinematically linear and nonlinear bar elements, subject to variation in the design parameter b200 and load magnitudes across the same range as used in Figs.[6.3.5-6.3.6].

Results

Most results were generated from physical parameters and loading conditions selec­ted to produce physically realistic values for local strains not exceeding 5%. The UL procedure, employed in this section for its compatibility with the more straightfor­ward 'intuitive' DSA approach, is less accurate than the TL procedure for localised strain levels in excess of 5%. A typical deformation, for P = 200 kips, is presented in Fig.(6.3.2] below.

Figure 6.3.2: Truss structure loading and deformation.

89

In Table[6.3.3] the comparative FDM and DSA stress sensitivities 88bun and objective

200 functional sensitivities ddwb

100, for variation in design parameter b200 , are presented for 200

elem. 199, the maximum strain element in the ti'uss, as a sampling of the 200 x 29 stress sensitivity vector generated by the implementation used here.

Physical Data Pr:= 200 kips Py= 200 kips b2oo = 0.1414 in 2

UL Analysis 0"11 = -568.077 ksi Uxl 74 = 1.581 in Uyi 74 = -2.025 in

Sens. procedure 8a11 d'11199

FDM/DSA --8b200 db200

FDM 93.505 ksi -2.656 DSA 89.616 ksi -2.545 104%

Table 6.:3.3: Comparative results sample.

\·Vi th reference to the introduction of this section, implementing multiple parameter DSA is crucial for large scale structural optimisation, as well as being non-trivial in its implementation into ABAQUS; Table[6.3.4] presents the 200 x 2 displacement sensivity vector ~b at node point 1 generated by the DSA procedure. In Figs.[6.3.3-6.3.4] a linear regression is performed for the FDM vs. DSA sensitivities generated for the element selection listed in the results summary.

FDM: Sensitivity of l.l! al ~ 0.2E-2 , ,

' I i

-0.2E-2

-0.4E-2

-0.6E-2 ··

-0.6E-2

-1.0E-2 '------''------'"---"'----'-----'-----'-----' -1.2E-2 -1.0E-2 -O.SE-2 -0.6E-2 -0.4E-2 -0.2E-2 O.OE-2 0.2E-2

DSA: Sensitivity of ~ at ~

d (1) Figure 6.3.3: Linear regression of FDM vs. DSA results for dub; •

200

FDM: Sensitivity of ~ at ~ 10.0E-3 ,

5.0E-3

-5.0E-3

-10.0E-3 ·-····· t·

-15.0E-3~-----------~---~--~

-15.0E-3 -10.0E-3 -5.0E-3 O.OE-3 5.0E-3 10.0E-3 DSA: Sensitivity of U.X at ~

d (1)

Figure 6.3.4: Linear regression of FDM vs. DSA results for dbuv • 200

90

'

92

In keeping with the ~pproach adopted in the preceding results sections, Figs.[6.3.5-6.3.6] and Figs.(6.3.7-6.3.8] respectively verify the stress-based DSA (vs. the FDM) and UL analysis (vs. ABAQUS) procedures for variation in the loadings Px and Py and the design parameter b200 . The maximum stress element 199 is monitored in both the DSA and UL analysis procedures, and the elem. 200 cross-section b200 is used consistently as a 'representative design parameter.

Sensitivity: Objective In. EID to b200 O.OE-3 .

-40.0E-3

-80.0E-3

FOM-OSA convergtnc1 1t t.3'4 attain In element 198

- DSA for e.iJ

'+

4.3t. •train In 111m1nt 19t

' ' '.

+· FDM for e.iJ ' \ ~----__..J E • 10000 ksl '-

-120.0E-3 L-------------------=~ 16 77 138 199 260

Loading magnitude ~. e.Jl (kips)

Figure 6.3.5: FDM and DSA sensitivities ddwb 199 for variation in Px. Py 200

Sensitivity: Objective In. EID to ~ 0.0 .---------=:__-==----------

E • 10000 ksi , loads: Px • 200 kips , Py• -200 k;ps

-0.4

-0.8

-1.2

0.141

5,7., strain in elem. 199

~ FDM-DSA convergence at 3.5 .. 1train In elem. 199

1.414

- DSA for ell

· +- FDM for ell

14.14 Cross-sectional design parameter l;!lQ.Q

Figure 6.3.6: FDM and DSA sensitivities ddwb 199 for variation in b200 200

91

DU DJ!(l} nu DBm DU DBm J2U DB(2) DU DB{l) DU DB<D DU lm{l} DU DB{2)

1 -0.414E-Ol ·0.97.5E-02 ? ·0.330E-02 -O.IJOE-02 3 0.4.57E-02 0.220E-02 4 0.148E-02 0.632E·03

5 0.393E-02 0.119E-Ol 6 ·0.203E·Ol 0.2.54E·Ol 7 0.127E-02 ·0.688E-OS 8 0.368E-03 0.4SlE-tU

' -0.186E-Ol -0.291E-02 10 -0.13.5E-02 -0.418E-03 1l -0.819E-03 0.12ZE-04 12 -0.712E-02 -0.264E-02

13 O.lOOE-02 0.38ZE-03 14 -0.418E·02 ·0.824E-03 15 -O.JSZE-03 -0.411E·03 1' 0.249E-02 0.96iE-03

17 -0.34SE-02 -0.133E-02 18 -0.421E·03 O.lOOE-02 19 -0.419E-02 0.971E-03 20 -O.SJOE-02 O.lOOE-OZ

21 -O.lllE-03 ·O.lllE-04 22 -0.726E-03 -0.844E-04 23 0.517E-03 0.108E·Ol 24 0.479E·Ol 0.104E-03

25 -O.JOlE-04 -0.666E-OS 26 0.190E·Ol O.S38E-Ol 27 -0.628E-02 -0.403E-OO 28 -0.193E-Ol 0.369E-OI

29 0.207E-02 0.236E-02 30 ·0.118E·Ol -0.117E-02 31 -0.303E·Ol -0.940E-03 32 ·0.2S2E·02 0.211E-05

33 ·0.404E·02 -0.109E·02 34 -0.143E-Ol -0.396E-02 35 -0.lOlE·Ol -0.200E-02 36 O.S46E-02 0.174E-02

37 -0.68SE-03 -0.349E·03 38 -0.819E-02 -0.JllE-02 39 -0 . .509E-02 -0.677E-02 40 -0.201E-02 0.681E-03

41 0.7S6E·03 -0.239E-03 42 O.lSJE-02 -0.449E-03 43 0.167E·Ol 0.228E·Ol 44 0.118E-02 0.270E-Ol

45 -0.130E-02 ·0.430E-03 46 O.S2.5E·02 0.140E-Ol 47 -0.424£-01 O.ZOOE-01 48 -0.740E-02 ·0.102E-02 49 -0.113E-Ol 0.264E-02 50 -0.460E-Ol -0.266E-02 51 O.S6SE-02 0.129E-02 52 ·0.280E·Ol -0.474E-02 53 ·0.210E-01 -0 . .58.5E·02 54 0.727E-02 0.184E-02 55 -0.363E-Ol -0.133E-Ol 56 ·0 . .529E-04 0.889E-03 57 ·0.877E-03 0.284E·0.5 58 -0.809E·03 0.808E-04 59 -O.USE-04 0 . .57SE-04 60 ·0.998E-OS 0.738E·04 61 -0.43ZE·03 -0.306E-03 62 -0.7.50E-04 ·0.221E·03 63 -0.123E-0.5 0.898E-07 64 0.404E-Ol 0.537E-Ol 65 -0.117E-Ol -0.342E-02 66 0.4S2E·02 O.JSlE-01 ,7 O.SUE-02 0.139E-01 68 ·0.209E-Ol -0.392E-02 69 ·O.SJOE-01 0.203E-Ol 70 -O.llJE-01 0.269E-02 71 -0.2S7E-02 -0.43SE-03 72 ·0.5.59E-Ol ·O.l38E-02 73 -0.409E-01 ·0.684E·02 74 0.107E-01 0.161E-02 75 ·0.187E·Ol -0.403E-02 76 ·O.SlSE·Ol ·0.187E-Ol 77 -0.667E-02 -0.468E·02 78 ·0.1S4E-03 -0.194E-03 79 ·O.USE-02 -0.703E-03 80 -0.179E·02 ·O.lSSE-02 81 0.496E-02 0.37.5E-02 82 -0.102E·Ol 0.673E-02 83 ·0.164E·Ol -0.306E-02 84 0.3.54E-02 0.337E-02 85 -0.492E·Ol 0.146E·Ol 86 ·0.373E·Ol ·0.1.54E-02 87 -0.281E·02 0.273E·02 88 -0 . .52iE-Ol 0.436E-02 89 ·0.230E-01 ·0.157E-02 98 ·0.174E-01 ·0.130E-02 91 -0.170E-Ol -0.720E-03 92 -0 . .520E-OZ -0.564E-OJ 93 -0.189E-01 -0.482E-02 94 0.290E-03 0.721E-03 95 -0.994E·03 -0.676E-03 " ·0.15.5E-OZ ·0.748E-OJ 97 ·0.104E-03 0.313E·04 98 0.603E-05 0.90ZE·0.5 " -0.986E-03 -O.llSE-03 100 -O . .SlZE-03 -0.109E-03 un 0 . .514E·0.5 ·O.llZE-04 102 0.922E-02 0.690E-02 103 -0.269E·Ol -O.ZOlE-02 104 -0.869E-02 0.128E·Ol 105 0.557E-02 0 . .534E-02 106 -0.634E-Ol -O . .SOIE-02 187 -0. 70ZE-Ol 0.136E-01 108 ·0.3.59E-02 0.343E-02 let -0.37ZE-01 -0.319E-02 110 -0.662E·Ol 0.348E-02 111 ·0.203E-01 -0.150E-02 112 0.147E·02 O.llOE-03 113 -0.120E·Ol ·0.816E-04 114 ·0.22ZE-Ol ·0 . .564E-02 11! -0.861E-02 -0.368E·02 116 ·0.206E-03 -0.198E-03 117 ·0.263E-02 0.276E-03 118 -0.lllE-02 -0.300E-03 119 0.474E-02 0.234E-02 120 ·0.114E-01 0.986E-02 121 -0.298E·Ol -0.493E-02 122 0.674E-02 0.37ZE-02 123 -0.88SE-01 0.140E-Ol 124 ·0.644E-01 -0.4.54E-02 125 ·0.349E-02 0 . .539E·02 126 -0.103 0 . .504E·02 127 -0.3.51E-01 ·0.169E-02 128 ·0.428E-Ol ·0.172E-02 129 ·0.21SE-Ol O.llZE-02 130 ·0.9.58E·03 0.414E-05 131 -0.361E·Ol ·0.644E-02 132 0.377E-03 0.473E-03 133 ·0.184E-02 -0.8.51E-03 134 ·0.244E-02 -0.918E-03 135 -0.211E-03 0.33.5E-03 136 0 . .574E-04 0.226E-03 137 -0.223E-02 0.142E-03 138 -0.217E-02 0.39ZE-04 139 ·0.178E-03 -O.S0.5E·04 140 0.107E-Ol 0 . .526E-02 141 ·0.36SE·Ol -0.266E-02 142 -0.419E-02 0.171E-Ol 143 0.104E-01 0 . .578E-02 144 -0.108 ·0.882E-02 145 -0.128 0.129E-Ol 146 -0.417E-02 0.630E·02 147 ·0 . .541E·Ol ·0.377E-02 148 -0.127 0.266E-02 149 ·0.47ZE·Ol -0.188E-02 150 0 . .512E·02 -0.388E-03 151 -0.lOZE·Ol 0.109E-02 152 -0.406E-Ol ·0.722E-02 153 ·0.171E·Ol ·0 . .54SE-02 154 0 . .516E-03 0.430E-03 155 -O.lOZE-01 0 . .530E·03 156 -0.916E-02 -0.918E-03 157 0.114E-02 0.417E-03 158 0.204E-OZ 0.114E-01 159 ·0.444E-Ol -0.627E-02 160 0 . .5.51E-02 0.201E-02 161 -0.161 O.lSOE-01 162 -0.971E·Ol -0.78SE-02 163 -0.28ZE-02 0.794E·02 164 -0.201 0.312E-02 165 -0.633E-01 -0.323E-02 166 -0.106 -0 . .512E-02 167 -0.24.5E-Ol 0.296E-02 168 0.3.SOE-02 -0.227E-03 169 -0.616E·Ol -0.770E-02 170 0.418E·03 0.341E-03 171 -0.434E·02 ·0.128E-02 172 ·O . .SlOE-02 . -0.139E-02 173 0.28.5E·03 0.60ZE-03 174 0 . .599E-03 0.384E-03 175 -0.633E·02 ·0.942E-05 176 -0.103E-Ol -0.619E-03 177 -0.1S.5E-02 ·0.229E-03 178 0.844E-02 0.30.5E-02 179 -0.380E·Ol -0.207E-02 180 0.189E-Ol 0.18.5E-Ol 181 0.117E·Ol 0.430E-OZ 182 -0.170 ·0.149E-01 183 -0.246 0.106E·Ol 184 -0.33SE-02 0.89.5E-02 185 ·0.103 -0. 71.5E-02 186 -0.266 ·0.248E-02 187 -0.113 ·0 . .$39E·02 188 -0.847E-02 -0.420E-02 189 -0.114E-02 0.21SE-02 190 -0.664E-Ol ·0.828E-02 191 -0.423E·02 -0.117E-02 192 -0.181E-Ol -0.3.51E·02 193 ·0.413E·Ol -0.176E-02 194 ·0.2.5.5E·Ol -0.18.5E-02 195 0.778E-02 0.249E-02 196 0.670E-02 0.220E-02 197 -0.186 -0.286E-Ol 198 ·0.3.54 ·0.209E-01 199 -~ -0.144 -0.706E-02 200 -0.114 -0.106E-01

Table 6.3.4: 200 parameter multivariate DSA.

Piola-Kirchholl (local) stress fill (ksi) Oi!!:-~~~~~~~~~~~~~~~~~~~--,

-100

-200

-300

O.S'Jlo strain

E • 10000 ksl , elem. 199

--€!- Updated Lagrange

· + · ABAQUS Nonlinear

-400 - ABAQUS Linear

0 40 80 120 160

Loading magnitude 12. ~ (kips) 200 240

Figure 6.3.7: Comparative stress results for variation in loads Px. Py.

-270

-320

-370

-420

-470

-520

Piela-Kirchhoff (local) stress fill (ksi)

E • 10000 kal , elem. 199

--€!- Updated Lagrange

· +· ABAOUS Nonlinear

- ABAQUS Linear

-51ou==---~~~~~~~~~~~~~~~~~~~-'

0.141 1.414 14.14 Cross-sectional design parameter b2.Q.Q

Figure 6.3.8: Comparative stress results for variation in b200 •

Results Discussion

93

• The linea.r regression in Figs.[6.3.3-6.3.4] .confirms the displacement-based 'in­tuitive' DSA results presented in Table[6.3:-it], in comparison with the FDM, for a selection of design parameters. The FDM and DSA results compare very well for the majority of paremeter sensitivities investigated; ie. FDM-DSA correspondence greater than 97% for realistic loading and deformation cases ( < 5% strain).

94

Notable exceptions tend to occur for

1. Elements under large strain (> 5%) ..

2. Very small values of sensitivity ~~;1 ( < I.OE - 3) . .

as was observed and commented upon in section (6.2).

• The deviation in the DSA results in case ( 1) are attributable to the failure of the UL procedure for latge strain, whereas case (2) is the result of numer­ical errors that occur due to a loss of significant numbers in the ABAQUS solver. The latter problem is largely irrelevant due to the negligible size of the sensitivities; in the former case, for large strains, a TL analysis proced­ure, in conjunction with the 'rigorous' DSA approach, is adopted to avoid . .

-maccurac1es.

• Figs.[6.3.5-6.3.6] present comparisons of the ('intuitive' approach) DSA vs. FDM sensitivities of the stress-based objective function, dd'Ilb 199

, for a range of 200

loads Px, Py and values of b200 , respectively.

In Fig.(6.3.5], the DSA and FDI\1 results predictably diverge for large values of strain. On the other hand, better than 99% correspondence is attained for strains of less than 1.3%.

The sigmoidal behaviour of the object function sensitivity, especially for small values of the design parameter b200 , is an accurate reflection of the actual physical behaviour of the truss: variations in the cross-sectional values b200,

for small values of b200 , result in roughly the same stress sensitivity response over a wide range of values b200 ; i.e., '11 199 becomes more sensitive to the other design parameters bi as b200 is decreased beyond a certain limit.

• Figs.(6.3. 7-6.3.8] verifies the UL kinematica.lly nonlinear element formulation employed here over the same ran'ge of variation in the loads Px, Py and design parameter b200 , as employed in Figs.[6.3.5-6.3.6]. ·

Conclusion

The results presented above demonstrate

~

• the suitability of the model in performing DSA for generalised objective and constraint functionals.

• that DSA can be successfully implemented in a commercial FE code, viz. ABAQUS, to conduct large parameter systems sensitivity analyses.

95

6.4 The 25-bar pylon truss,

Introduction

The pylon truss structure investigated here is representative of a whole range of 3D structural problems commonly encountered in industry. Three dimensional struc­tural optimisation problems frequently arise in fields as diverse as electrificatio.Q and off-shore oil production {22].

The results presented in this section seek to demonstrate the following two vital extensions to the subject matter of the preceding results sections:

J. The implementation of the 'rigorous' displacement-based non-shape DSA ap­proach utilising the TL analysis formulation for large strain problems.

IJ. The implementation of a shape (geometric) DSA procedure, employing the 'rigorous' sensitivity analysis approach.

z 2.54

x

)--,

2.54

5.08

Figure 6.4.1: 25-bar pylon configuration.

96

The particular functional sensitivity vector formulations for the two cases above, as well as information pertaining to the structural design and loading of the pylon, are summarised in Table(6.4.l].

Results Summary

The results section is divided into two parts, respectively verifying the two DSA enhancements listed above:

1. Graphical results for a representative loading and deformation of the pylon truss structure is presented in Fig.(6.4.2].

2. Fig.(6.4.3] provides a diagrammatic distinction between the shape and non­shape DSA approaches adopted in the investigation of the pylon's structural behaviour; that is, giving physical meaning to the material and geometric design parameters, b4 and h, respectively.

3. Table(6.4.2] contains comparative FDM and DSA functional sensitivities, d~~3

> 1Ji(3)

and d db4

, for representative loading Pz and design parameter h values.

4. In Figs.(6.4.4-6.4.9] the displacement responses, in the y and z directions at node point 3, of the TL, UL, ABAQUS linear and ABAQUS nonlinear formu­lations are compared for variation in the non-shape design parameter, b4, the shape design parameter , h, and the vertical load, Pz, respectively.

5 A . f 1 . . . . dll'(J) d dll'(J) d d b ,. t 't . companson o t 1e sens1t1v1ty estimates,~ an ~'pro uce y m UI -

ive' (with UL) and 'rigorous' (with TL) DSA implementations, as well as the FDM, for variation in b4 and Pz, is presented in Figs.(6.4.10-6.4.13].

6. In introducing the shape DSA formulation results, Figs.(6.4.14] presents the

functional sensitivity d~t for a limited selection ~f structural configurations_.

7. Figs.(6.4.15-6.4.16] compares the FDM and 'rigorous' shape DSA functional • • • • dll'(3) cl dllt(3) .f f t . d . t sens1tiv1ty estimates, Tan T' 1or a range o geome rzc es1gn parame er

values h and loads Pz.

•

3

L~

Figure 6.4.2: Initial and Deformed truss structure.

Results

The diagrams in Fig.[6.4.3] lend physical meaning to the sample shape, h, and non­shape, b4, design parameters employed in the sensitivity analysis of the 25-bar pylon structure.

z

x

)--,

- -----· , , , , .· ,

, •• •••• ,,' ---- - --:-, 4 •• ·: ,,'

... , , -~ .·'th ,

. ':·: _ ~ -_:'.: _ :~:: ~~~~:~~~:~~: ~ ~ ~:: .. _ ::: . -----Shape design parameter 'h'

Non-shape design parameter 'b~

Figure 6.4.3: Shape and Non-shape design parameters.

97

- Young's modulus - Element cross-section - Structure proportions

- Load

- Constraint

98

Physical properties

E = 6.9 x 107 kPa bi = A = 3.225 x 10-<j m2 , i = 1 ... 25 cf. Fig.[6.4.1)

Loads and Constraints

• Node point 3 - p(3) = ( 1.92, 19.2, -Pz ) Node point 4 - p(4

) = (-1.92, -19.2, -Pz ) where Pz = 240 kN, as illustrated in Fig.(6.4.1]. Node points 7, 8, 9 and 10 are fixed.

Analysis procedure

- TL and UL kinematically nonlinear

DSA procedure

- Non-shape sensitivities Vb w~i) calculated w.r.t. cross-sectional design parameters b, utilising a displacement-based 'rigorous' DSA approach. (j)

- Cakulate d~ in order to obtain the objective functional sensitivity V'b w!il, where

w~i) = (uV>) - 1 so that i I '

- Shape DSA sensitivities Vh w~il calculated with regard to the geometric design parameter h, also utilising the 'rigorous' DSA approach.

(j)

- The shape parameter displacement sensitivities ~;i , and hence the objective functional

sensitivities v h w~il, are found in a fashion analogous to the procedure employed for obtaining Vb wlil, so that

'7 w\il = dw i = (2 uV>) dui V h I - dh I dh

(j) ( (j))

with j = node number and i = y or z

Table 6.4.1: Analysis data.

99

In order to estimate the relative accuracy of the 'rigorous' DSA approach, used in conjunction with the TL analysis procedure, Table[6.4.2] provides a comparison of FDM and 'rigorous' DSA results for shape and· non-shape functional sensitivities, subject to a representative selection of load and boundary conditions (cf. Fig.(6.4.1 ]). Additionally, results generated by the 'intuitive' DSA approach have also been in­cluded for non-shape sensitivity analysis. As mentioned in chapter 3, shape, or geo­metric, sensitivity analysis is unfeasible with the 'intuitive' DSA approach, hence the omission of results for DSA1.

The results predictably indicate a relatively high displacement sensitivity, in node point 3, to variation in the cross-sectional design parameter b4 for the given struc­tural configuration; sensitivity to variation in the geometric design parameter h is

(3)

negligible. An increased shape sensitivity d~~ for large values his demonstrated in Fig.[6.4.16].

Additionally, from Table[6.4.2] it may be observed that, for the small strain deform­ation case examined, the DSAi results perform marginally better than DSAr relative to the FDM.

Non-shape Shape

d\!J(3) dw<3J d\!J(3) d\!J(3) _ Y_ z _Y _ z

db4 db4 dh dh

OS Ar -0.717778E-1 -0.976143 0.544858E-6 0.816130E-3 OS Ai -0.703918E-1 -0.955257 - -FDM -0.673398E-1 -0.932395 0.510149E-6 0.818099E-3

DSAr/FDM 106.6% 104.7% 106.8% 99.8%

bi=3.225x10-4 m2 with i=l ... 25, Pz=240 kN . E = 6.9xl07 kPa I max. Exxl = 0.41 %

DSAi : 'intuitive' approach , DSAr : 'rigorous' approach.

Table 6.4.2: DSA vs. FDM - Comparing~sensitivities at node 3

100

The displacement results in Figs.[6.4..1-6.:1.9] a.re generated for variation in the design parameters and loads. The results verify the UL and TL analysis procedures, ap­plied to the pylon truss structure, by comparison with the ABAQUS kinematically nonlinear element formulation. The displacement results are furthermore analysed in order to gain additional insight into the relationship between displacement and functional sensitivity, especially in the case of the 'intuitive' DSA approach.

Due to the tendency of the pylon structure to distribute the load imposed through as many elements as possible, large deformation occurs in conjunction with relat­ively small values of strain. Numerical problems (specifically bifurcation behaviour) prevent imposing material parameters and/or loads which would generate strains in the pylon truss structure in excess of 1 %. A comparison of the maximum strain in any element of the pylon structure(~ 1%) with the maximum strains occuring in the 3-bar truss (cf. Fig.[6.1.3]) enables a proper assessment of the reduced range of behaviour observable for the pylon structure.

Consequently, most displacement results w~re generated for relatively small values of strain €xx and, hence, good correspondence between the various nonlinear formu­lations is maintained. Only for strain values exceeding 3%-4% would divergence of the TL and UL values become more apparent.

Additionally, as will be demonstrated in the case of non-shape design parameter sensitivity for small strain, any variation between TL and UL based results is more likely to be attributable to numerical error than to inherent characteristics of the formulations.

Displacement: !h at node ;!

·2.0E-3

-4.0E-3

-6.0E-3

·8.0E-3

bl • 3.225E-4 aqr. m. , I • 1...3,5 ... 25

+ Updated Lagrange: U:;:

-+- Total Lagrange: U:;:

-El- ABAOUS Nonlinear: U:;:

""*- ABAOUS Linear: U:;:

-10.0E-3'---------------------1.000E-04 1.000E-03 ~

Cross-sectional design parameter !2.4 0.01

Figure 6.4.4: Displacements u~3 ) vs. cross-section b4 .

-2.oe-1

-4.0E-1

-6.0E-1

-6.0E-1

-10.oe-1

max. strain • 0.55'11. compreaaive at elm. 25

max. strain: 1.39'11. compressive in elem. 22

E • 6.9E•7 kPa , h • 0

- OSA(r) tor !eJ..iLx · + · OSA(i) for !eJ..iLx -B- FOM for !fJ..iLx bl • 3.225E·4 aqr. m.. i • 1. •. 25

-12.oe-1~~~~~~~~~~~~~~~~~~~~~~

8 107 206 305 404 503 602 Vertical load fZ (kN)

701 800

Figure 6.4.12: dw (3)

Non-shape sensitivity db: vs. load Pz.

Sensitivity: ~ to ll! 0.01!fil--==e!I=:::::-::==---==-----------~

max. strain: 1.39'11. compressive In elem. 22

-2.0

-4.0

max. strain • 0.55'11. compressive at elm 25

-6.0

-8.0 E • 6.9E•7 kPa, h • 0

OSA(r) for le.J.il.j

-10.0 + · OSA(i) for !flil..:

-B- F 0 M lo r !flil..: bl • 3.225E-4 aqr. m., I • 1 .•• 25

-12.0L..~~~~~~~~~~~~~~~~~~~~~~

8 107 206 305 404 503 602 701 800 Vertical load ~ (kN)

(11lJ(3) Figure 6.4.13: Non-shape sensitivity db: vs. load Pz.

105

Displacement: ·~ at node ~

-2.4E-2 E • 6.9E•7 kPa, Pz • 240 kN. h • 0

-2.9E-2

-3.4E-2

bl • 3.225E-4 sqr. m. , I • 1. •• 3,5 ••• 25

~ Updated Lagrange: ILi:

· + Total Lagrange: ILi:

· 0- ABAOUS Nonllnear: U%

""*- ABAQUS Linear: U%

-3.9E-2L-~~~~~~~~~~~~~~~~~~~~--'

1.000E-04 1.000E-03 Cross-sectional design parameter b4

0.01

Figure 6.4.5: Displacements u~3l vs. cross-section b4 •

Displacement: !J.y at node ;3 -2.3E-3~~~~~~~~~~~~~~~~~~~~~--.

-2.SE-3

-2.6E-3

-2.7E-3

-1.0

Pz • 240 kN Updated Lagrange: U.X

-+- Total Lagrange: U.X - El- ABAQUS Nonlinear: U:t - X- ABAOUS Linear: U.X

bl• 3.225E·4 aqr. m •• I • 1. •• 25

Max. strain: 0.84'!1. compreaalve In elem. 22

0.0 1.0 2.0 Shape design parameter b

Figure 6.4.6: Displacements ut~) vs. shape design h.

101

-3.5E-3

-5.5E-3

-7.5E-3

-9.5E-3

Displacement: !,!z at node 3

E • 6.9E•7 kPa Pz • 240,kN

Updated Lagrange: IU

+- Total Lagrange: UJ

i3 · ABAQUS Nonlinear: UJ

--*"" ABAOUS Linear: UJ

bl • 3.225E·4 sqr. m. , I • 1...25

Max. strain: 0.64"' compre11lve In elem. 22

-1.0 0.0 1.0 Shape design parameter !:!

2.0

Figure 6.4. 7: Displacements u13) vs. shape design h.

Displacement: !J..x at node 3 O.OE-3...,...-'------'-----------------,

-2.0E-3

-4.0E-3

-6.0E-3

-8.0E-3

8 107 206 305

Updated Lagrange: Ux

--l- Total Lagrange: la

-Er ABAQUS Nonlinear: Ux -*- ABAOUS Linear: Ux

bl • 3.225E-4 aqr. m. , I • 1 ••• 25

E • 6.9E• 7 kPa , h • 0 m

404 503 602 701

Vertical load ~ (kN)

Figure 6.4.8: Displacements uil vs. load Pz.

800

102

Displacement: !.!.; at node ~ o.oe-2---------------------.

-2.0E-2

-4.0E-2

-6.0E-2

-8.0E-2

- Updated Lagrange: UI

· + Al!IAOUS Linear: UI

· 0 ABAQUS Nonlinear: UI

~ Total Lagrange: UI

E • 6.9E•7 kPa , h • 0 m

-10.0E-2 ~-------------------'t­. 8 107 206 305 404 503 602 701 800

Vertical load ~ (kN)

Figure 6.4.9: Displacements uP) vs. load Pz.

103

The 'rigorous' DSA approach (with TL) is adopted for shape sensitivity analysis due to the infeasibility of performing shape DSA with the 'intuitive' approach (with UL). For non-shape sensitivity analysis of a structure subject to large deformation, however, the main purpose in utilising the 'rigorous' approach stems from the de­teriorating accuracy of the UL formulation for large deformation accompanied by large strain.

As was the case for the displacement results, the large deformation/ small strain behaviour of the pylon structure causes convergence of the 'rigorous' and 'intuit­ive' DSA formulations for a wide range of variation in the cross-sectional design parameter b4 (cf. Figs.[6.4.10-6.4.11]).

In Figs.[6.4.12-6.4.13], however, distinct divergence between the various sensitivity estimates is clearly observable for relatively small strain values ( ~ 0.5% ). More importantly, the results apparently suggest that the 'intuitive' DSA approach (DSAi) produces more accurate sensitivity results than the 'rigorous' DSA approach (DSAr ), in direct contradiction to the theory-based expectations. This anomality is discussed towards the end of the section.

Sensitivity: ~ to M

-2.0E-1

-4.0E-1

-6.0E-1

-8.0E-1

-10.oe-1

-12.0E-1

Pz • 240 kN , h • O

bl • 3.225E·4 aqr. m., l • 1. .. 3,5 ... 25

· -0- · DSA(r) for !fall-l! -+- DSA(i) for !fall-l! -e- FD M for !fall-l!

·14.0E-1"1=-~~~~~~~~~~~~~~~~~~~--'

1.000E-04 1.000E-03 Cross-sectional design parameter M

0.01

Figure 6.4.10: d\ll (3)

Non-shape sensitivity db: vs. cross-section b4.

Sensitivity: ~ to M

-2

-4

-e

1.000E,04

E • 6.9E•7 kPa , Pz • 240 kN , h • 0

bi • 3.225E·4 aqr. m., I • 1. •• 3,5 ... 25

max. strain • 0.94'1!o compressive at element 4

~ DSA(r) for {fail_:

+ · DSA(I) for {fail_z

-e- FDM for {fail_z

1.000E-03 Cross-sectional design parameter M

0.01

• dW.(3)

Figure 6.4.11: Non:--shape sensitivity db: vs. cross-section b4.

104

106

As demonstrated in Table(6.4.2], the structural configuration in Fig.[6.4.1] exhibits negligible sensitivity to variation in h, whereas for an incrementally enlarged pylon base the structure becomes progressively more sensitive to 6.h, cf. Fig.[6.4.14].

From the table in Fig.f6.4.14] it is evident that a 400% increase in the pylon base

l · 1a r ld · · l · · · dw~3> area resu ts m an - o mcrease m be sens1t1v1ty ~·

h 0.00 1.23 2.46

dllJ(3) z 0.816E-3 0.377E-2 0.144E-1 dh

bi = 3.225x10-4 m2 with i = 1 ... 25 Pz = 240 kN , E = 6.9xl07 kPa

h =0.00 9

F. 6 S I f. . d h . . . dw ~3 > igure .4.14: tructura con rguratron an s ape sens1t1v1ty --;n;-·

107

The results in Figs.(6.4.15-6.4.16] pertain to the second part of the subject matter dealt with in this section, that is the enhancement of the sensitivity formulation so as to deal with shape sensitivity analysis. Very. close correspondence between the FDM and the shape DSA procedures is observed in the results. The close corres­pondence is at least partially attributable to the greater accuracy of the FDM for small sensitivities, whereas for larger sensitivities divergence between the FDM and the DSA implementations is in no small part due to the increasing inaccuracy of the FDM as a numerical technique.

Figs.(6.4.15-6.4.16] illustrate that for the structural configuration in Fig.(6.4.15] (h = 0), the pylon response is markedly insensitive to small variations in the pylon base area, implying a very stable design. Fig.(6.4.14] demonstrates decreasing stability with increasing base area.

Sensitivity: ~ to !l Sensitivity: !e.fill.: to !l 8.7E-7+-----------------,:

E • 6.9E•7 kPa , Pz • UO .kN

7. 7E-7 \ bi • 3.225E-• aqr. m., I • 1. •• 25

6.7E-7

-1.0 0.0 1.0. Shape design parameter !l

- DSA for te.sil..:t -<>- DSA for !.f.ail...z

·+· FDM for te.sil..:t ·-><·· FDM for !.f.ail...z

2.0

12.0E-3

9.0E-3

6.0E-3

dw(3) dw(3 )

Figure 6.4.15: Shape sensitivities di: , dh vs. design parameter h.

6.0E-6

5.0E-6

4.0E-6

3.0E-6

2.0E·6

1.0E-6

Sensitivity: !.e.ru1..y to !J

+ OSAlor~

+· FOMlor~ -&- CSA for !.elil...i ·:<· FOM for !.elil...i

Sensitivity:~ to !J

E • 6.9E•7 kPa • h • 0

bl • 3.225E·4 sqr. m .. I• 1 ... 25

O.OE-64"-~'----------------1

8.0E-3

6;0E-3

4.0E-3

2.0E-3

8 107 206 305 404 503 602 701 800

Vertical load Pz (kN)

d\.If (3) d\.If (3) Figure 6.4.16: Shape sensitivities d~ , dh vs. load Pz.

Results discussion

108

In the results presented in this section for the structural and sensitivity analyses of the pylon truss structure, the 'rigorous' (TL based) and shape DSA procedures were investigated as enhancements to the procedures utilised in the preceding sections.

A comparative analysis of the enhanced procedures verified their correctness and underscored their usefulness viz. the implementations of the preceding sections. In what follows below, certain peculiarities observed in the results are analysed and discussed.

• As mentioned above, due to the tendency of the pylon to deform with minimum strain, clear divergence between the UL and TL based formulations - a result of the UL formulation's inability to accurately model large strain behaviour - is not clearly observable. To be more rrecise: large deformation is accompanied by small strain; attempting to deform the structure sufficiently in order to produce large strains more often leads to numerical instability.

• A consequence of adopting a limited range of strain values to investigate the structural and sensitivity analysis models is the disproportionate impact of numerical errors on the results: the more complex the formulation of a model (eg. TL or DSAr), the more prone it becomes to machine errors; eg. loss of significance, truncation and rounding. Numerical errors have a negligible impact on the results when assessed over.a wide range of strain values, as in the case of the 3-bar truss (cf. Figs.[6.1.3-6:1.4]). For the results presented in Figs.[6.4.4-6.4.5] and Figs.[6.4.12-6.4.13], however, the small strain range and numerical errors are responsible for some, usually unapparent, anomalous behaviour; eg. the apparent reduced accuracy of the TL or DSAr formulations. Over a larger strain range, the intrinsic characteristics of the various models dominate inaccuracies due to numerical error; ie. for large strain, TL-based models outperform UL-based formulations.

109

6.5 Optimising the 25-bar pylon truss

Introduction

Two distinct PLBA structural optimisation procedures are applied to the 25-bar pylon structure in order to find the optimal values of, respectively, single shape and non-shape design parameters.

The pylon structure is preferred for the purposes of structural optimisation due to its relatively simple design, stable behaviour and general appeal. Pylon and off-shore structures provide ideal practical real-world benchmarks: the designs are standard­ised and data for these structures, subject to various types of loading conditions, is readily available from the literature.

The results presented here aim

• To verify the successful implementation of the PLBA structural optimisation, thereby concluding one of the main undertakings of this thesis, namely the inclusion of optimisation into an established commercial FEA package.

• To demonstrate the successful application of structural optimisation to shape and non-shape optimal design problems.

Data pertaining to the structural design, material parameters and loading/boundary conditions for the pylon structure are similar to that of section 6.4 (cf. Table[6.4.l]).

Results Summary

Three sets of results are provided in this section:

• The first set of results verifies the performance of the PLBA RQP optimisation routine separately. The optimal spring design problem presented in [26] is investigated. The problem involves four primary inequality constraints and three side-constraints, cf. Table[6.5.l].

• The second set of optimisation results s~eks to demonstrate the applicability of ABAQUS to non-shape structural optimisation. A single generic cross­sectional design parameter is optimised, cf. Table[6.5.3].

• The final set of optimisation results demonstrates the applicability of ABAQUS to shape structural optimisation: a single shape design parameter h2

, propor­tional to the base area of the pylon structure, is optimised, cf. Table[6.5.5].

2Refer to Fig.[6.4.l]

110

Results

The results for each of the three optimisation sets enumerated above are presented in Table(6.5.2], Table(6.5.4] and Table(6.5.6], respectively.

- Initial Design

- Design Parameters

- Cost Function (minimise)

- Constraint Functions

Problem Specification

d = wire diameter = 1 D = mean coil diameter= 2 n = number active coils = 3

d, D, n

f=(n+2)Dd2

- - D3n < 0 91 - 1 718J5d4 -

_ ( 4D - dD) 1 _ 1

< O 92

- (12566(Dd3 - d4 )) + (5108d2 ) -

- - 140.45d < 0 93 - 1 (D2n) -

= (D + d) -1<0 94 1.5 -95 = 0.05 - d :::; 0 96 = 0.05 - D :::; 0 91=1 - n:::; 0

Table 6.5.1: Spring optimisation: verifying the PLBA procedure.

Results

Final Cost Final Design f d D ~n NI NF

0.0128 0.0538 0.409 10.7 46 254

Table 6.5.2: Spring optimisation: verifying the PLBA procedure.

1 2 3

Problem Specification

- Design and Loads - Design Parameter Definition - DSA Procedure

cf. Table[6.4.1) cf. Figf 6.4.3) 'Rigorous' Non-shape sensitivity

NEL

- Cost Function f = L L(i)A(i), total pylon volume i=l

(minimise)

- Constraint Functions (

(3) ) 2 - uz -1 < 0 91 - 2.5 x 10-2 -

(

u(3) ) 2

- y -1<0 92 - 2.0 x 10-3 -

Table 6.5.3: Optimising the pylon truss w.r.t. cross-section b4 •

Results

Initial Design Initial Cost Final Design Final Cost NI

l.OOOxl0-4 m 2 3.00x10-2 m 3 6.239 x 10-3 m 2 4.68x10-2 m 3 64 3.225x 10-4 m 2 3.06x10-2 m 3 5.914x 10-3 m2 4.59x10-2 m 3 51 l.OOOxl0-3 m 2 3.25 x 10-2 m 3 5.830 x'10::::3 m 2 4.56x10-2 m 3 27

Table 6.5.4: Optimising the pylon truss w.r.t. cross-section b4 •

111

NF.

159 127 71

112

Problem Specification

- Design and Loads - Design Parameter Definition

cf. Table[6.4.l], with p(4)= (-1.92, 0.0, -240 kN) cf. Fig[6.4.3]

- DSA Procedure 'Rigorous' Shape sensitivity

- Cost Function (

(5) ) 2

j = 1.0:ylQ-4 - 1 :5 0

(minimise)

- Constraint Function uz

(

(3) ) 2 . - . -1 < . 91 - 5.0xlQ-3. - 0

Table 6.5.5: Optimising the pylon truss w.r.t. base area index h.

Results

Initial Design Initial Cost Final Design Final Cost NI NF

1 -0.025 2.33 x 10-3 m3 0.482 9. 72x10-5 m3 22 39 2 0.000 7.31x10-4 m3 0.493 9.86x10-5 m3 20 33 3 0.025 l.92x 10-4 m3 0.485~ l.14x 10-4 m3 17 26

Table 6.5.6: Optimising the pylon truss w.r.t. base area index h.

113

Results discussion

This section largely represents the culmination, .or verification, of the research effort undertaken in this thesis.

In most cases the rates of convergence were comparable to those presented in the literature for similar problems. For both shape and non-shape structural optimisa­tion relatively stable structural configurations were selected as initial designs - the PLBA optimisation procedure was not subjected to extreme scenarios comparable to what was used to test the structural and sensitivity analysis models. This ap­proach is justified as the primary focus of this research is concentrated on sensitivity analysis. Investigating the overall robustness of the optimisation model is a future undertaking.

The results presented here are not directly comparable to similar examples inves­tigated by other authors, e.g. Haug and Arora[15]. This is largely a consequence of the fact that the results presented here are based on an analysis model which neglects the effect of body force, or self-weight, on the structure.

Various minor difficulties were encountered during the implementation of the diffe­rent sets of optimisation problems:

• Occasionally, especially for shape optimisation, the design sensitivity coeffi­cients (DSC) in a particular sensitivity vector became sufficiently small to force termination of the DSA procedure, thereby providing the optimisation module with spurious results.

• An initially unrobust line search routine, applied to the descent function, on occasion exhibited instability thereby adversely affecting the optimisation pro­cedure.

• Substantial overhead code in the SIMPLEX procedure, used to solve the linear­ised QP problem, resulted in a noticeable decrease in computational efficiency.

An important component of future research will entail a further refinement of the optimisation procedures employed.

Chapter 7

Conclusion

7.1 Research Achievements

The numerical results presented in Chapter 6 verified the successful completion of the primary objectives of this thesis1

. To extrapolate from Chapter 6, the achievements of this work can be summarised as follows:

I. The formulation of design sensitivity analysis (DSA), specifically the direct differentiation method (DDM), in 'pseudo'-element form, thereby theoretic­ally enabling the incorporation of sensitivity analysis in any established finite element analysis (FEA) code.

II. The implementation into ABAQUS of a range of different DSA procedures, specifically:

(1) The 'intuitive' non-shape DSA approach, here utilising the UL kinematic model.

(2) The 'rigorous' non-shape DSA approach, utilising the TL kinematic model.

(3) Shape design sensitivity analysis.

Additionally, results were presented to demonstrate:

• Multi-parameter DSA (cf. Section 6.3~Table(6 .. 3.4]).

• The feasibility of non-displacement based functional sensitivity analysis (cf. Section 6.3 Table(6.3.3], Figs.[6.3.5-6.3.6]).

• The influence of large strain on the behaviour of selected DSA formula­tions (cf. Section 6.1).

1 Refer to Summary.

114

115

III. The implementation of structural optimisation in conjunction with an estab­lished commercial FEA code, for instance ABAQUS, which enables the struc­tural optimisation of problems too complex to be attempted by small scale, focussed optimisation routines.

7 .2 Results Discussion

Many of the observations and comments made for the specific problems implemented and analysed in Chapter 6 may be further generalised. In addition, many of the issues or results highlighted in the discussion concluding each results section are worthwhile emphasising.

7.2.1 Design Sensitivity Analysis

Small Strain

The uncertainty, or error margin, of the small strain (2·3% max. strain £xx) shape and non-shape DSA 2 results, as compared to the FDM, is less than 5% for the largest sensitivities3

. Considering that the accuracy of the FDM fluctuates significantly de­pending on the specific application, the DSA behaviour may be even better than the comparison suggests. Points of interest in the behaviour of the various formulations for small strain are:

• The results obtained indicate a particularly good comparison between the DSA and FDM results for shape sensitivity analysis (cf. Fig.[6.4.15]).

• As a rough rule of thumb, small sensitivities are a characteristic consequence of small strain behaviour4

• It is important to note that small sensitivities are particularly susceptable to machine error (truncation, rounding) due to peculiarities in the ABAQUS implementation. This occasionally results in a sizeable divergence between the DSA and FDM small strain estimates (cf. Table[6.2.3]). This effect, though noticeable, is generally disregarded due to the negligible size of the sensitivities inv?lved.

• Due to the complexity of the 'rigorous' DSA implementation, machine error dominates for very small strain, occasionally resulting in a noticeable diver­gence when compared to results produced using the 'intuitive' approach (cf.

2 'intuitive' or 'rigorous', UL or TL. 3 Refer to Tables(6.l.2], [6.2.3], [6.3.4], [6.4.2] 4 Notable exceptions exist.

116

Figs[6.4.12-6.4.13]). This behaviour is overcome for large strain where the 'intuitive' approach becomes progressively more inaccurate.

Large Strain

Noticeable variation occurs in the performance of the various DSA approaches for large strain, as indicated and discussed in the numerical results section. Additional aspects of the DSA large strain behaviour that should be emphasised are:

• For the purposes of the work presented in this thesis the 'intuitive' DSA ap­proach utilises the Updated Lagrange (UL) formulation5 and, consequently, exhibits decreasing accuracy for large strain. The TL formulation's accurate performance for large strain is the primary motivation for it being employed, in preference to the UL, in the 'rigorous' DSA approach (cf. Table[6.2.2], Figs.[6.2. 7-6.2.8]). Figs.[6.2.5-6.2.6] in Section 6.2 suggest the possible failure of the UL formulation for very large strain.

• A particular shortcoming of the Section 6.4 was the omission of numerical results representing the behaviour of the various DSA procedures for large strain. As mentioned previously, this is due to the tendency of the pylon structure to distribute loads efficiently and so deform with minimal strain. Overloading the structure in order to observe large strain behaviour results in numerical instability.

7 .2.2 Structural Optimisation

The successful implementation of the PLBA RQP optimisation procedure into a commercial FEA package, viz. ABAQUS, fulfills a fundamental objective of this research in addition to consolidating the various DSA formulations utilised.

Structural optimisation was performed for relatively simple constraint and objective function sets. For these, and a range of additional simulations performed, the opti­misation routine enabled a successful investigation of the problem of finding optimal shape and non-shape design parameters for the 25-bar pylon structure.

Results converge at rate comparable to that p;esented in the literature for similar constraint and objective function scenarios. Specific assumptions made in this work, eg. neglecting body force in the equilibrium formulation, preclude direct comparison

5In this research the 'intuitive' method has consistently been associated with the UL variational formulation and the 'rigorous' approach with the TL. The 'intuitive' formulation can, however, also be used in conjunction with the TL; likewise the UL variational approach could be used in conjunction with the 'rigorous' approach.

117

with similar examples in the literature. Further research will involve the implemen­tation of additional optimisation models, eg. the truss region method [32], enabling the evaluation of the efficiency of the optimisation procedure employed here.

7.3 Future Research

Wide scope exists for extending the research undertaken in this thesis - the range of topical issues in DSA and structural optimisation are extensive. The main objectives of most research conducted in the field of design sensitivity analysis, particularly when applied to structural optimisation, are increased accuracy and computational speed. Future research effort will be focussed primarily on extending the current research in the following areas:

• The kinematic model may be improved by choosing a more efficient element formulation based on the enhanced assumed strain variational description. This would offer a cost efficient opportunity to implement structural opti­misation involving higher order elements. One such possible application al­lows for material parameter optimisation for sheet metalforming: the interface between ABAQUS, DSA and structural optimisation developed in this thesis enables the utilisation ABAQUS' sophisticated solving, contact and friction procedures for the purposes of structural optimisation.

• A secondary goal involves implementing design sensitivity analysis, and even­tually structural optimisation, for a nonlinear material model. Work in this area is fraught with difficulty - even the simplest kinematic hardening proced­ures are seemingly intractable.

For the sake of comparison various other non-DDM based sensitivity approaches and optimisation procedures could be investigated. For the purposes of this research such a wide approach was considered to be beyond the scope of the objectives set for this thesis.

, '

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